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+arXiv:2301.08614v1  [math.AP]  20 Jan 2023
+Plane wave stability analysis of Hartree and quantum
+dissipative systems
+Thierry Goudon∗1 and Simona Rota Nodari†1
+1Université Côte d’Azur, Inria, CNRS, LJAD,
+Parc Valrose, F-06108 Nice, France
+Abstract
+We investigate the stability of plane wave solutions of equations describing quantum
+particles interacting with a complex environment. The models take the form of PDE
+systems with a non local (in space or in space and time) self-consistent potential; such
+a coupling lead to challenging issues compared to the usual non linear Schrödinger
+equations. The analysis relies on the identification of suitable Hamiltonian structures
+and Lyapounov functionals. We point out analogies and differences between the original
+model, involving a coupling with a wave equation, and its asymptotic counterpart
+obtained in the large wave speed regime. In particular, while the analogies provide
+interesting intuitions, our analysis shows that it is illusory to obtain results on the
+former based on a perturbative analysis from the latter.
+Keywords. Hartree equation. Open quantum systems. Particles interacting with a vibrational
+field. Schrödinger-Wave equation. Plane wave. Orbital stability.
+Math. Subject Classification. 35Q40 35Q51 35Q55
+1
+Introduction
+This work is concerned with the stability analysis of certain solutions of the following Hartree-type
+equation
+iBtU ` 1
+2∆xU “ γ
+ˆ
+σ1 ‹x
+ˆ
+Rn σ2Ψ dz
+˙
+U,
+(1a)
+´ ∆zΨ “ ´γσ2pzq
+`
+σ1 ‹x |U|2˘
+pxq
+(1b)
+∗thierry.goudon@inria.fr
+†simona.rotanodari@univ-cotedazur.fr
+1
+
+endowed with the initial condition
+U
+ˇˇ
+t“0 “ UInit,
+(2)
+and of the following Schrödinger-Wave system:
+iBtU ` 1
+2∆xU “ γΦU,
+(3a)
+1
+c2 B2
+ttΨ ´ ∆zΨ “ ´γσ2pzqσ1 ‹ |U|2pt, xq,
+(3b)
+Φpt, xq “
+¨
+TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dz dy,
+(3c)
+where γ, c ą 0 are given positive parameters, completed with
+U
+ˇˇ
+t“0 “ UInit,
+Ψ
+ˇˇ
+t“0 “ ΨInit,
+BtΨ
+ˇˇ
+t“0 “ ΠInit.
+(4)
+The variable x lies in the torus Td, meaning that the equations are understood with p2πq´periodicity
+in all directions. In (3b), the additional variable z lies in Rn and, as explained below, it is crucial to
+assume n ě 3. For reader’s convenience, the scaling of the equation is fully detailed in Appendix A;
+for our purposes the God-given form functions σ1, σ2 are fixed once for all and the features of
+the coupling are embodied in the parameters γ, c. The system (1a)-(1b) can be obtained, at least
+formally, from (3a)-(3c) by letting the parameter c run to `8, while γ is kept fixed. By the way,
+system (1a)-(1b) can be cast in the more usual form
+iBtU ` 1
+2∆xU “ ´γ2κ
+`
+Σ ‹x |U|2˘
+U,
+t P R, x P Rd.
+(5)
+where1
+κ “
+ˆ
+Rn σ2pzqp´∆zq´1σ2pzq dz “
+ˆ
+Rn
+|pσ2pξq|2
+|ξ|2
+dξ
+p2πqn ą 0 and Σ “ σ1 ‹ σ1.
+(6)
+Letting now Σ resemble the delta-Dirac mass, the asymptotic leads to the standard cubic non linear
+Schrödinger equation
+iBtU ` 1
+2∆xU “ ´γ2κ|U|2U.
+(7)
+in the focusing case. These asymptotic connections can be expected to shed some light on the
+dynamics of (3a)-(3c) and to be helpful to guide the intuition about the behavior of the solutions,
+see [20, 21].
+The motivation for investigating these systems takes its roots in the general landscape of the
+analysis of “open systems”, describing the dynamics of particles driven by momentum and energy
+exchanges with a complex environment. Such problems are modeled as Hamiltonian systems, and
+it is expected that the interaction mechanisms ultimately produce the dissipation of the particles’
+energy, an idea which dates back to A. O. Caldeira and A. J. Leggett [7]. These issues have been
+investigated for various classical and quantum couplings, and with many different mathematical
+viewpoints, see e. g. [2, 3, 24, 25, 28, 29, 30]. The case in which the environment is described as
+a vibrational field, like in the definition of the potential by (3b)-(3c), is particularly appealing. In
+1The Fourier transform of an integrable function ϕ : Rn Ñ C is defined by pϕpξq “
+´
+Rn ϕpzqe´iξ¨z dz.
+2
+
+fact, (3a)-(3c) is a quantum version of a model introduced by S. De Bièvre and L. Bruneau, dealing
+with a single classical particle [6]. Intuitively, the model of [6] can be thought of as if in each space
+position x P Rd there is a membrane oscillating in a direction z P Rn, transverse to the motion
+of the particles. When a particle hits a membrane, its kinetic energy activates vibrations and the
+energy is evacuated at infinity in the z´direction. These energy transfer mechanisms eventually
+act as a sort of friction force on the particle, an intuition rigorously justified in [6, Theorem 2 and
+Theorem 4]. We refer the reader to [1, 12, 13, 30, 46] for further theoretical and numerical insight
+about this model. The model of [6] has been revisited by considering many interacting particles,
+which leads to Vlasov-type equations, still coupled to a wave equation for defining the potential
+[17]. Unexpectedly, asymptotic arguments indicate a connection with the attractive Vlasov-Poisson
+dynamic [11]. In turn, the particles-environment interaction can be interpreted in terms of Lan-
+dau damping [19, 18]. The quantum version (3a)-(3c) of the De Bièvre-Bruneau model has been
+discussed in [21, 20], with a connection to the kinetic model by means of a semi-classical analysis
+inspired from [35]. Note that in (3a)-(3c), the vibrational field remains of classical nature; a fully
+quantum framework is dealt with in [3] for instance.
+A remarkable feature of these systems is the presence of conserved quantities, here inherited
+from the framework designed in [6] for a classical particle, and the study of these models brings out
+the critical role of the wave speed c ą 0 and the dimension n of the space for the wave equation
+(we can already notice that n ě 3 is necessary for (6) to be meaningful), see [6, 18, 19, 21]. For the
+Schrödinger-Wave system (3a)-(3c) the energy
+HSWpU, Ψ, Πq “ 1
+4
+ˆ
+Td |∇U|2 dx ` 1
+4
+¨
+TdˆRn
+ˆΠ2
+c2 ` |∇zΨ|2
+˙
+dx dz ` γ
+2
+ˆ
+Td Φ|U|2 dx,
+(8)
+is conserved since we can readily check that
+d
+dtHSWpU, Ψ, BtΨq “ 0.
+Similarly, for the Hartree system (1a)-(1b), we get
+d
+dtHHapUq “ 0
+where we have set
+HHapUq “ 1
+4
+ˆ
+Td |∇U|2 dx ´ γ2 κ
+4
+ˆ
+Td Σpx ´ yq|Upt, xq|2|Upt, yq|2 dy dx.
+Furthermore, for both model, the L2 norm is conserved. Of course, these conservation properties
+play a central role for the analysis of the equations. However, (1a)-(1b) has further fundamental
+properties which occur only for the asymptotic model: firstly, (1a)-(1b) is Galilean invariant, which
+means that, given a solution pt, xq ÞÑ upt, xq and for any p0 P Td, the function pt, xq ÞÑ upt, x ´
+tp0qeipx´tp0{2q is a solution too; secondly, the momentum pptq “ Im
+´
+¯upt, xq∇xupt, xq dx is conserved
+and, accordingly, the center of mass follows a straight line at constant speed. That these properties
+are not satisfied by the more complex system (3a)-(3c) makes its analysis more challenging. Finally,
+we point out that, in contrast to the usual nonlinear Schrödinger equation or Hartree-Newton
+system, where Σ is the Newtonian potential, the equations (1a)-(1b) or (3a)-(3c) do not fulfil a
+3
+
+scale invariance property. This also leads to specific mathematical difficulties: despite the possible
+regularity of Σ, many results and approaches of the Newton case do not extend to a general kernel,
+due to the lack of scale invariance.
+When the problem is set on the whole space Rd, one is interested in the stability of solitary
+waves, which are solutions of the equation with the specific form upt, xq “ eiωtQpxq, and, for
+(3a)-(3c), ψpt, x, zq “ Ψpx, zq. The details of the solitary wave are embodied into the Choquard
+equation, satisfied by the profile Q, [32, 36]. It turns out that the Choquard equation have infinitely
+many solutions; among these solutions, it is relevant to select the solitary wave which minimizes the
+energy functional under a mass constraint, [32, 37] and to study the orbital stability of this minimal
+energy state. This program has been investigated for (7) and (1a)-(1b) in the specific case where
+Σpxq “
+1
+|x| in dimension d “ 3, by various approaches [8, 31, 33, 34, 39, 49, 50]. Quite surprisingly,
+the specific form of the potential plays a critical role in the analysis (either through explicit formula
+or through scale invariance properties), and dealing with a general convolution kernel, as smooth
+as it is, leads to new difficulties, that can be treated by a perturbative argument, see [27, 51] for
+the case of the Yukawa potential, and [21] for (1a)-(1b) and (3a)-(3c).
+Here, we adopt a different viewpoint. We consider the case where the problem holds on the
+torus Td, and we are specifically interested in the stability of plane wave solutions of (3a)-(3c) and
+(1a)-(1b). We refer the reader to [4, 5, 14, 40] for results on the nonlinear Schrödinger equation
+(7) in this framework. The discussion on the stability of these plane wave solutions will make the
+following smallness condition
+4γ2κ}σ1}2
+L1 ă 1
+(9)
+(assuming the plane wave has an amplitude unity) appear. Despite its restriction to the periodic
+framework, the interest of this study is two-fold: on the one hand, it points out some difficulties
+specific to the coupling and provides useful hints for future works; on the other hand, it clarify the
+role of the parameters, by making stability conditions explicit.
+The paper is organized as follows. In Section 2, we clarify the positioning of the paper. To
+this end, we further discuss some mathematical features of the model. We also introduce the main
+assumptions on the parameters that will be used throughout the paper and we provide an overview
+of the results. Section 3 is concerned with the stability analysis of the Hartree equation (1a)-(1b).
+Section 4 deals with the Schrödinger-Wave system at the price of restricting to the case where the
+wave vector of the plane wave solution vanishes: k “ 0. For reasons explained in details below, the
+general case is much more difficult. Section 5 justifies that in general the mode k � 0 is linearly
+unstable. Finally, in Appendix A, we provide a physical interpretation of the parameters involved,
+and for the sake of completeness, in Appendices B and C, we discuss the well-posedness of the
+Schrödinger-Wave system (3a)-(3c) and its link with the Hartree equation (1a)-(1b) in the regime
+of large c’s.
+4
+
+2
+Set up of the framework
+2.1
+Plane wave solutions and dispersion relation
+For any k P Zd, we start by seeking solutions to (3a)-(3c) of the form
+Upt, xq “ Ukpt, xq :“ exp
+`
+ipωt ` k ¨ xq
+˘
+,
+Ψpt, x, zq “ Ψ˚pzq,
+BtΨpt, x, zq “ Π˚pzq “ 0,
+(10)
+with ω ě 0. Note that the L2 norm of Uk is p2πqd{2 and Ψ˚ actually does not depend on the time
+variable, nor on x. Since |Ukpt, xq| “ 1 is constant, the wave equation simplifies to
+1
+c2 B2
+ttΨ ´ ∆zΨ “ ´γσ2pzq
+@
+σ1
+D
+Td,
+where
+@
+¨
+D
+Td stands for the average over Td:
+@
+f
+D
+Td “
+´
+Td fpxq dx. As a consequence, z ÞÑ Ψ˚pzq is
+a solution to (3b) if
+Ψ˚pzq “ ´γΓpzq
+@
+σ1
+D
+Td,
+with Γ the solution of
+´∆zΓpzq “ σ2pzq.
+This auxiliary function Γ is thus defined by the convolution of σ2 with the elementary solution of
+the Laplace operator in dimension n, or equivalently by means of Fourier transform:
+Γpzq “
+ˆ
+Rn
+Cn
+|z ´ z1|n´2 σ2pz1q dz1 “ F ´1
+ξÑz
+´pσ2pξq
+|ξ|2
+¯
+.
+(11)
+The corresponding potential (3c) is actually a constant which reads
+´γ
+¨
+TdˆRn σ1px ´ yqσ2pzqΓpzq
+@
+σ1
+D
+Td dz dy “ ´κγ
+@
+σ1
+D2
+Td
+with
+κ “
+ˆ
+Rn σ2pzqΓpzq dz “
+ˆ
+Rn |∇zΓpzq|2 dz ą 0
+(we remind the reader that this formula coincides with (6) and makes sense only when n ě 3). It
+remains to identify the condition on the coefficients so that Uk satisfies the Schrödinger equation
+(3a): this leads to the following dispersion relation
+ω ` k2
+2 ´ Υ˚ “ 0,
+Υ˚ “ γ2κ
+@
+σ1
+D2
+Td ą 0
+(12)
+with k2 “ řd
+j“1 k2
+j. We can compute explicitly the associated energy:
+HSWpUk, Ψ˚, Π˚q “ p2πqd
+2
+ˆk2
+2 ´ γ2κ
+2
+@
+σ1
+D2
+Td
+˙
+“ p2πqd
+4
+pk2 ´ Υ˚q.
+Of course, among these solutions, the constant mode U0pt, xq “ eiωt1pxq has minimal energy.
+It turns out that the plane wave Ukpt, xq “ eiωteik¨x equally satisfies (1a)-(1b) provided the
+dispersion relation (12) holds. Incidentally, we can check that
+HHapUkq “ p2πqd
+2
+ˆk2
+2 ´ γ2κ
+2
+@
+Σ
+D
+Td
+˙
+“ p2πqd
+4
+pk2 ´ Υ˚q
+is made minimal when k “ 0.
+5
+
+2.2
+Hamiltonian structure and symmetries of the problem
+The conservation properties play a central role in the stability analysis, for instance in the reasonings
+that use concentration-compactness arguments [8]. Based on the conserved quantities, one can try
+to construct a Lyapounov functional, intended to evaluate how far a solution is from an equilibrium
+state. Then the stability analysis relies on the ability to prove a coercivity estimate on the variations
+of the Lyapounov functional, see [47, 49, 50]. This viewpoint can be further extended by identifying
+analogies with finite dimensional Hamiltonian systems with symmetries, which has permitted to
+set up a quite general framework [22, 23], revisited recently in [4]. The strategy relies on the ability
+in exhibiting a Hamiltonian formulation of the problem
+BtX “ JBXH pXq,
+where the symplectic structure is given by the skew-symmetric operator J. As a consequence of
+Noether’s Theorem, this formulation encodes the conservation properties of the system. In partic-
+ular, it implies that t ÞÑ H pXptqq is a conserved quantity. For the problem under consideration, as
+it will be detailed below, X is a vectorial unknown with components possibly depending on different
+variables (x P Td and z P Rn). This induces specific difficulties, in particular because the nature
+of the coupling is non local and delicate spectral issues arise related to the essential spectrum of
+the wave equation in Rn. Next, we can easily observe that the systems (1a)-(1b) and (3a)-(3c)
+are invariant under multiplications by a phase factor of U, the “Schödinger unknown”, and under
+translations in the x variable. This leads to the conservation of the L2 norm of U and of the total
+momentum. However, the systems (1a)-(1b) and (3a)-(3c) cannot be handled by a direct applica-
+tion of the results in [4, 22, 23]: the basic assumptions are simply not satisfied. Nevertheless, our
+approach is strongly inspired from [4, 22, 23]. As we will see later, for the Hartree system, a decisive
+advantage comes from the conservation of the total momentum and the Galilean invariance of the
+problem. For the Schrödinger-Wave problem, since the expression of the total momentum mixes
+up contribution from the “Schrödinger unknown” U and the “wave unknown” Ψ, the information
+on its conservation does not seem readily useful. 2
+In what follows, we find advantages in changing the unknown by writing Upt, xq “ eik¨xupt, xq;
+in turn the Schrödinger equation iBtU ` 1
+2∆U “ ΦU becomes
+iBtU ` 1
+2∆u ´ k2
+2 u ` ik ¨ ∇u “ Φu.
+Accordingly, the parameter k will appear in the definition the energy functional H . This explains
+a major difference between (1a)-(1b) and (3a)-(3c): for the former, a coercivity estimate can be
+obtained for the energy functional H , for the latter, when k � 0 there are terms which cannot be
+controlled easily. This is reminiscent of the momentum conservation in (1a)-(1b) and the lack of
+Galilean invariance for (3a)-(3c). The detailed analysis of the linearized operators sheds more light
+on the different behaviors of the systems (1a)-(1b) and (3a)-(3c).
+2For the problem set on Rd, it is still possible, in the spirit of results obtained in [14] for NLS, to justify
+that orbital stability holds on a finite time interval: the solution remains at a distance ǫ from the orbit of
+the ground state over time interval of order Op1{ ?ǫq, see [48, Theorem 4.2.11 & Section 4.6]. The argument
+relies on the dispersive properties of the wave equation through Strichartz’ estimates.
+6
+
+2.3
+Outline of the main results
+Let us collect the assumptions on the form functions σ1 and σ2 that govern the coupling:
+(H1) σ1 : Td Ñ r0, 8q is C8 smooth, radially symmetric;
+@
+σ1
+D
+Td � 0;
+(H2) σ2 : Rn Ñ r0, 8q is C8 smooth, radially symmetric and compactly supported;
+(H3) p´∆q´1{2σ2 P L2pRnq;
+(H4) for any ξ P Rn, pσ2pξq � 0.
+Assumptions (H1)-(H2) are natural in the framework introduced in [6]. Hypothesis (H3) can
+equivalently be rephrased as p´∆q´1σ2 P .H1pRnq; it appears in many places of the analysis of
+such coupled systems and, at least, it makes the constant κ in (6) meaningful. This constant is
+a component of the stability constraint (9). Hypothesis (H4) equally appeared in [6, Eq. (W)]
+when discussing large time asymptotic issues. Assumptions (H1)-(H4) are assumed throughout
+the paper.
+Our results can be summarized as follows. We assume (9) and consider k P Zd and ω ą 0
+satisfying (12). For the Hartree equation, the analysis is quite complete:
+• the plane wave eipωt`k¨xq is spectrally stable (Theorem 3.1);
+• for any initial perturbation with zero mean, the solutions of the linearized Hartree equation
+are L2-bounded, uniformly over t ě 0 (Theorem 3.3);
+• the plane wave eipωt`k¨xq is orbitally stable (Theorem 3.5).
+For the Schrödinger-Wave system, only the case k “ 0 is fully addressed:
+• the plane wave peiωt1pxq, ´γΓpzq
+@
+σ
+D
+Td, 0q is spectrally stable (Corollary 5.12);
+• for any initial perturbation of peiωt1pxq, ´γΓpzq
+@
+σ
+D
+Td, 0q with zero mean, the solutions of the
+linearized Schrödinger-Wave system are L2-bounded, uniformly over t ě 0 (Theorem 4.2);
+• the plane wave peiωt1pxq, ´γΓpzq
+@
+σ
+D
+Td, 0q is orbitally stable (Theorem 4.4).
+When k � 0, the situation is much more involved; at least we prove that in general the plane wave
+solution peipωt`k¨xq, ´γΓpzq
+@
+σ1
+D
+Td, 0q is spectrally unstable, see Section 5 and Corollary 5.15.
+3
+Stability analysis of the Hartree system (1a)-(1b)
+To study the stability of the plane wave solutions of the Hartree system, it is useful to write the
+solutions of (1a)-(1b) in the form
+Upt, xq “ eik¨xupt, xq
+with upt, xq solution to
+iBtu ` 1
+2∆u ´ k2
+2 u ` ik ¨ ∇u “ ´γ2κpΣ ‹ |u|2qu.
+(13)
+7
+
+If k P Zd and ω ą 0 satisfy the dispersion relation (12), uωpt, xq “ eiωt1pxq is a solution to (13) with
+initial condition uωp0, tq “ 1pxq. Therefore, studying the stability properties of Ukpt, xq “ eiωteik¨x
+as a solution to (1a)-(1b) amounts to studying the stability of uωpt, xq “ eiωt1pxq as a solution to
+(13).
+The problem (13) has an Hamiltonian symplectic structure when considered on the real Banach
+space H1pTd; Rq ˆ H1pTd; Rq. Indeed, if we write u “ q ` ip, with p, q real-valued, we obtain
+Bt
+ˆ
+q
+p
+˙
+“ J∇pq,pqH pq, pq
+with
+J “
+ˆ 0
+1
+´1
+0
+˙
+and
+H pq, pq “ 1
+2
+ˆ1
+2
+ˆ
+Td |∇q|2 ` |∇p|2 dx ` k2
+2
+ˆ
+Tdpp2 ` q2q dx ´
+ˆ
+Td pk ¨ ∇q dx `
+ˆ
+Td qk ¨ ∇p dx
+˙
+´ γ2κ
+4
+ˆ
+Td Σ ‹ pp2 ` q2qpp2 ` q2q dx.
+Coming back to u “ q ` ip, we can write
+H puq “ 1
+2
+ˆ1
+2
+ˆ
+Td |∇u|2 dx ` k2
+2
+ˆ
+Td |upxq|2 dx `
+ˆ
+Td k ¨ p´i∇uqu dx
+˙
+´ γ2κ
+4
+ˆ
+TdpΣ ‹ |u|2qpxq|upxq|2 dx.
+(14)
+As observed above, H is a constant of the motion.
+Moreover, it is clear that (13) is invariant under multiplications by a phase factor so that
+Fpuq “ 1
+2}u}2
+L2 is conserved by the dynamics. The quantities
+Gjpuq “ 1
+2
+ˆ
+Td
+ˆ1
+i Bxju
+˙
+u dx
+are constants of the motion too, that correspond to the invariance under translations. Indeed, a
+direct verification leads to
+d
+dtGjpuqptq “ κγ2
+2
+ˆ
+Td
+ˆ
+Td BxjΣpx ´ yq ‹ |u|2pt, yq|u|2pt, xq dy dx “ 0.
+Finally, we shall endow the Banach space H1pTd; Rq ˆ H1pTd; Rq with the inner product
+Bˆ
+q
+p
+˙ ˇˇˇ
+ˆ
+q1
+p1
+˙F
+“
+ˆ
+Td
+`
+pp1 ` qq1q dx.
+that can be also interpreted as an inner product for complex-valued functions:
+xu|u1y “ Re
+ˆ
+Td uu1 dx.
+(15)
+8
+
+3.1
+Linearized problem and spectral stability
+Let us expand the solution of (13) around uω as upt, xq “ uωpt, xqp1 ` wpt, xqq. The linearized
+equation for the fluctuation reads
+iBtw ` 1
+2∆xw ` ik ¨ ∇xw “ ´2γ2κpΣ ‹ Repwqq.
+(16)
+We split w “ q ` ip, q “ Repwq, p “ Impwq so that (16) recasts as
+Bt
+ˆq
+p
+˙
+“ Lk
+ˆq
+p
+˙
+(17)
+with the linear operator
+Lk :
+ˆ
+q
+p
+˙
+ÞÝÑ
+¨
+˝
+´k ¨ ∇xq ´ 1
+2∆xp
+1
+2∆xq ` 2γ2κΣ ‹ q ´ k ¨ ∇xp
+˛
+‚.
+(18)
+Theorem 3.1 (Spectral stability for the Hartree equation) Let k P Zd and ω ą 0 such that
+the dispersion relation (12) is satisfied. Suppose (9) holds. Then the spectrum of Lk, the lineariza-
+tion of (13) around the plane wave uωpt, xq “ eiωt1pxq, in L2pTd; Cq ˆ L2pTd; Cq is contained in iR.
+Consequently, this wave is spectrally stable in L2pTdq.
+Proof.
+To prove Theorem 3.1, we expand q, p and σ1 by means of their Fourier series
+qpt, xq “
+ÿ
+mPZd
+Qmptqeim¨x,
+Qmptq “
+1
+p2πqd
+ˆ
+Td qpt, xqe´im¨x dx,
+ppt, xq “
+ÿ
+mPZd
+Pmptqeim¨x,
+Pmptq “
+1
+p2πqd
+ˆ
+Td ppt, xqe´im¨x dx,
+σ1pxq “
+ÿ
+mPZd
+σ1,meim¨x,
+σ1,mptq “
+1
+p2πqd
+ˆ
+Td σ1pxqe´im¨x dx.
+Note that σ1 being real and radially symmetric, we have
+σ1,m “ σ1,m “ σ1,´m
+(19)
+and, by definition,
+@
+σ1
+D
+Td “ p2πqdσ1,0. As a consequence, we obtain
+Lk
+ˆq
+p
+˙
+“
+¨
+˚
+˚
+˝
+ř
+mPZd
+ˆm2
+2 Pm ´ ik ¨ mQm
+˙
+eim¨x
+ř
+mPZd
+ˆ
+´m2
+2 Qm ´ ik ¨ mPm ` 2p2πq2dγ2κ|σ1,m|2Qm
+˙
+eim¨x
+˛
+‹‹‚
+“ Lk,0
+ˆQ0
+P0
+˙
+`
+ÿ
+mPZd∖t0u
+Lk,m
+ˆQm
+Pm
+˙
+eik¨x
+(20)
+with
+Lk,0 “
+ˆ
+0
+0
+2p2πq2dγ2κ|σ1,0|2
+0
+˙
+and Lk,m “
+˜
+´ik ¨ m
+m2
+2
+´ m2
+2 ` 2p2πq2dγ2κ|σ1,m|2
+´ik ¨ m
+¸
+(21)
+for m P Zd ∖ t0u.
+9
+
+Note that, since the Fourier modes are uncoupled,
+ˆq
+p
+˙
+is a solution to (17) if and only if the
+Fourier coefficients
+ˆQm
+Pm
+˙
+satisfy
+Bt
+ˆ
+Qmptq
+Pmptq
+˙
+“ Lk,m
+ˆ
+Qmptq
+Pmptq
+˙
+for any m P Zd. Similarly, λ P C is an eigenvalue of the operator Lk if and only if there exists at
+least one Fourier mode m P Zd such that λ is an eigenvalue of the matrix Lk,m, i.e. there exists
+pqm, pmq � p0, 0q such that
+λqm ´ m2
+2 pm ` ik ¨ mqm “ 0,
+λpm ` m2
+2 qm ` ik ¨ mpm “ 2p2πq2dγ2κ|σ1,m|2qm.
+(22)
+A straightforward computation gives that λ0 “ 0 is the unique eigenvalue of the matrix Lk,0
+with eigenvector p0, 1q. This means that KerpLkq contains at least the vector subspace spanned by
+the constant function x P Td ÞÑ
+ˆ0
+1
+˙
+, which corresponds to the constant solution upt, xq “ i of (16).
+Next, if m P Zd ∖ t0u, λm is an eigenvalue of Lk,m if it is a solution to
+pλ ` ik ¨ mq2 ´ m2
+2
+ˆ
+´m2
+2 ` 2p2πq2dγ2κ|σ1,m|2
+˙
+“ 0.
+This is a second order polynomial equation for λ and the roots are given by
+λm,˘ “ ´ik ¨ m ˘ |m|
+2
+b
+´m2 ` 4γ2κp2πq2d|σ1,m|2.
+If the smallness condition (9) holds, the argument of the square root is negative for any m P Zd∖t0u,
+and thus the roots λ are all purely imaginary (and we note that λ´m,˘ “ λm,¯). More precisely,
+we have the following statement.
+Lemma 3.2 (Spectral stability for the Hartree equation) Let k, m P Zd and Lk,m defined
+as in (21). Then
+1. λ0 “ 0 is the unique eigenvalue of Lk,0 and KerpLk,0q “ span
+"ˆ
+0
+1
+˙*
+;
+2. for any m P Zd ∖ t0u, the eigenvalue of Lk,m are
+λm,˘ “ ´ik ¨ m ˘ |m|
+2
+b
+´m2 ` 4γ2κp2πq2d|σ1,m|2.
+(a) if 4γ2κp2πq2d |σ1,m|2
+m2
+ď 1, then λm,˘ P iR;
+(b) if 4γ2κp2πq2d |σ1,m|2
+m2
+ą 1, then λm,˘ P C ∖ iR. Moreover, Repλm,`q ą 0.
+Now, (9) implies 4γ2κp2πq2d |σ1,m|2
+m2
+ă 1 for all m P Zd ∖ t0u, so that σpLkq Ă iR and uωpt, xq “
+eiωt1pxq is spectrally stable. Conversely, if σ1, σ2 and γ are such that there exists m˚ P Zd ∖ t0u
+10
+
+verifying 4γ2κp2πq2d |σ1,m˚|2
+m2
+˚
+ą 1, then the plane wave uω is spectrally unstable for any k P Zd and
+ω ą 0 that satisfy the dispersion relation (12). This proves Proposition 3.1.
+We observe that this result is consistent with the linear stability analysis of (7), see [40, The-
+orem 1], when replacing formally Σ by the delta-Dirac. The analogy should be considered with
+caution, though, since the functional difficulties are substantially different: here u ÞÑ ´ 1
+2∆Tdu ´
+2γ2κΣ‹Repuq is a compact perturbation of ´ 1
+2∆Td, which has a compact resolvent hence a spectral
+decomposition.
+It is important to remark that the analysis of eigenproblems for Lk has consequences on the
+behavior of solutions to (17) of the particular form
+Qpt, xq “ eλtqpxq,
+Ppt, xq “ eλtppxq.
+We warn the reader that spectral stability excludes the exponential growth of the solutions of the
+linearized problem when the smallness condition (9) holds, but a slower growth is still possible.
+This can be seen by direct inspection for the mode m “ 0: we have BtQ0 “ 0, so that Q0ptq “ Q0p0q
+and BtP0 “ 2p2πq2dκ
+@
+σ1
+D2
+TdQ0p0q which shows that the solution can grow linearly in time
+P0ptq “ P0p0q ` 2p2πq2dγ2κ
+@
+σ1
+D2
+TdQ0p0qt.
+In fact, excluding the mode m “ 0 suffices to guaranty the linearized stability.
+Theorem 3.3 (Linearized stability for the Hartree equation) Suppose (9).
+Let w be the
+solution of (16) associated to an initial data wInit P H1pTdq such that
+´
+Td wInit dx “ 0.
+Then,
+there exists a constant C ą 0 such that suptě0 }wpt, ¨q}H1 ď C.
+Proof.
+Note that if
+´
+Td wInit dx “ 0 then the corresponding Fourier coefficients Q0p0q and P0p0q
+are equal to 0. As a consequence, Q0ptq “ P0ptq “ 0 for all t ě 0, so that
+´
+Td wpt, xq dx “ 0 for all
+t ě 0.
+The proof follows from energetic consideration. Indeed, we observe that, on the one hand,
+1
+2
+d
+dt
+ˆ
+Td |∇w|2 dx “ ´γ2κ
+2i
+ˆ
+Td Σ ‹ pw ` wq∆pw ´ wq dx,
+and, on the other hand,
+1
+2
+d
+dt
+ˆ
+Td Σ ‹ pw ` wqpw ` wq dx
+“ ´ 1
+2i
+ˆ
+Td Σ ‹ pw ` wq∆pw ´ wq dx ´ k ¨
+ˆ
+Td ∇pw ` wqΣ ‹ pw ` wq dx,
+where we get rid of the last term in the right hand side by assuming k “ 0. This leads to the
+following energy conservation property
+d
+dt
+"1
+2
+ˆ
+Td |∇w|2 dx ´ γ2κ
+2
+ˆ
+Td Σ ‹ pw ` wqpw ` wq dx
+*
+“ 0
+which holds for k “ 0. We denote by E0 the energy of the initial data wInit. Finally, we can simply
+estimate
+ˇˇˇˇ
+ˆ
+Td Σ ‹ pw ` wqpw ` wq dx
+ˇˇˇˇ ď }Σ ‹ pw ` wq}L2}w ` w}L2 ď }Σ}L1}w ` w}2
+L2 ď 4}Σ}L1}w}2
+L2.
+11
+
+To conclude, we use the Poincaré-Wirtinger estimate. Indeed, since we have already remarked that
+the condition
+´
+Td wInit dx “ 0 implies
+´
+Td wpt, xq dx “ 0 for any t ě 0, we can write
+}wpt, ¨q}2
+L2 “
+›››wpt, ¨q ´
+1
+p2πqd
+ˆ
+Td wpt, yq dy
+›››
+2
+L2 “ p2πqd
+ÿ
+mPZd∖t0u
+|cmpwpt, ¨qq|2
+ď p2πqd
+ÿ
+mPZd∖t0u
+m2|cmpwpt, ¨qq|2 “ }∇wpt, ¨q}2
+L2
+for any t ě 0, where the cmpwpt, ¨qq’s are the Fourier coefficients of the function x P Td ÞÑ wpt, xq.
+Hence, for any solution with zero mean, we infer, for all t ě 0,
+2E0 “
+ˆ
+Td |∇w|2pt, xq dx´γ2κ
+ˆ
+Td Σ‹pw `wqpw `wqpt, xq dx ě p1´4γ2κ}Σ}L1q
+ˆ
+Td |∇wpt, xq|2 dx.
+As a consequence, if (9) is satisfied, we obtain
+sup
+tě0
+}wpt, ¨q}H1 ď 2
+d
+E0
+1 ´ 4γ2κ}Σ}L1 .
+The stability estimate extends to the situation where k � 0. Indeed, from the solution w of
+(16), we set
+vpt, xq “ wpt, x ` tkq.
+It satisfies iBtv ` 1
+2∆xv “ ´2γ2κΣ ‹ Repvq. Hence, repeating the previous argument, }vpt, ¨q}H1 “
+}wpt, ¨q}H1 remains uniformly bounded on p0, 8q. This step of the proof relies on the Galilean invari-
+ance of (5); it could have been used from the beginning, but it does not apply for the Schrödinger-
+Wave system.
+Remark 3.4 The analysis applies mutadis mutandis to any equation of the form (1a), with the
+potential defined by a kernel Σ and a strength encoded by the constant γ2κ. Then, the stability
+criterion is set on the quantity 4γ2κp2πqd |pΣm|
+m2 For instance, the elementary solution of pa2´∆xqΣ “
+δx“0 with periodic boundary condition has its Fourier coefficients given by pΣm “
+1
+p2πqdpa2`m2q ą 0.
+Coming back to the physical variable, in the one-dimension case, the function Σ reads
+Σpxq “ e´a|x|
+2a
+`
+coshpaxq
+ape2aπ ´ 1q.
+The linearized stability thus holds provided 4γ2κp2πq2d
+1
+a2`1 ă 1.
+3.2
+Orbital stability
+In this subsection, we wish to establish the orbital stability of the plane wave uωpt, xq “ eiωt1pxq
+as a solution to (13) for k P Zd and ω ą 0 that satisfy the dispersion relation (12). As pointed
+out before, (13) is invariant under multiplications by a phase factor.
+This leads to define the
+corresponding orbit through upxq “ 1pxq by
+O1 “ teiθ, θ P Ru.
+12
+
+Intuitively, orbital stability means that the solutions of (13) associated to initial data close enough
+to the constant function x P Td ÞÑ 1 “ 1pxq remain at a close distance to the set O1. Stability
+analysis then amounts to the construction of a suitable Lyapounov functional satisfying a coercivity
+property. This functional should be a constant of the motion and be invariant under the action of
+the group that generates the orbit O1. Hence, the construction of such a functional relies on the
+invariants of the equation. Moreover, the plane wave has to be a critical point on the Lyapounov
+functional so that the coercivity can be deduced from the properties of its second variation. The
+difficulty here is that, in general, the bilinear symmetric form defining the second variation of the
+Lyapounov function is not positive on the whole space: according to the strategy designed in [22],
+see also the review [47], it will be enough to prove the coercivity on an appropiate subspace. Here
+and below, we adopt the framework presented in [4] (see also [5]).
+Inspired by the strategy designed in [4, Section 8 & 9], we introduce, for any k P Zd and ω ą 0
+satisfying the dispersion relation (12), the set
+Sω “
+!
+u P H1pTd; Cq, Fpuq “ Fp1q “ p2πqd
+2
+“ p2πqd k2{2 ` ω
+2γ2κ
+@
+σ1
+D2
+Td
+)
+;
+Sω is therefore the level set of the solutions of (13), associated to the plane wave pt, xq ÞÑ uωpt, xq “
+eiωt1pxq. Next, we introduce the functional
+Lωpuq “ H puq ` ωFpuq ´
+dÿ
+j“1
+kjGjpuq,
+(23)
+which is conserved by the solutions of (13). We have
+BuLωpuqpvq
+“
+Re
+ˆ1
+2
+ˆ
+Tdp´∆uqv dx ` k2
+2
+ˆ
+Td uv dx
+´γ2κ
+¨
+TdˆTd Σpx ´ yq|upyq|2upxqvpxq dy dx`ω
+ˆ
+Td uv dx
+˙
+.
+As a matter of fact, we observe that
+BuLωp1q “ 0
+owing to the dispersion relation. Next, we get
+B2
+uLωpuqpv, wq
+“
+Re
+ˆ1
+2
+ˆ
+Tdp´∆ ` k2qwv dx
+´2γ2κ
+¨
+TdˆTd Σpx ´ yqRe
+`
+upyqwpyq
+˘
+upxqvpxq dy dx
+´γ2κ
+¨
+TdˆTd Σpx ´ yq|upyq|2wpxqvpxq dy dx ` ω
+ˆ
+Td wv dx
+˙
+.
+Still by using the dispersion relation, we obtain
+B2
+uLωp1qpv, wq “ Re
+¨
+˚
+˚
+˚
+˝
+ˆ
+Td
+ˆ
+´∆w
+2
+´ 2γ2κΣ ‹ Repwq
+˙
+looooooooooooooooomooooooooooooooooon
+:“Sw
+vpxq dx
+˛
+‹‹‹‚“ xSw|vy.
+13
+
+S : H2pTdq Ă L2pTdq Ñ L2pTdq is an unbounded linear operator and its spectral properties will
+play an important role for the orbital stability of uω. Note that the operator S is the linearized
+operator (18), up to the advection term k ¨ ∇.
+The main result of this subsection is the following.
+Theorem 3.5 (Orbital stability for the Hartree equation) Let k P Zd and ω ą 0 such that
+the dispersion relation (12) is satisfied. Suppose (9) holds. Then the plane wave uωpt, xq “ eiωt1pxq
+is orbitally stable, i.e.
+@ε ą 0, Dδ ą 0, @vInit P H1pTd; Cq, }vInit ´ 1}H1 ă δ ñ sup
+tě0
+distpvptq, O1q ă ε
+(24)
+where distpv, O1q “ infθPr0,2πr }v ´ eiθ1} and pt, xq ÞÑ vpt, xq P C0pr0, 8q; H1pTdqq stands for the
+solution of (13) with Cauchy data vInit.
+The key ingredient to prove Theorem 3.5 is the following coercivity estimate on the Lyapounov
+functional.
+Lemma 3.6 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose that
+there exist η ą 0 and c ą 0 such that
+@w P Sω, dpw, O1q ă η ñ Lωpwq ´ Lωp1q ě c distpw, O1q2.
+(25)
+Then the the plane wave uωpt, xq “ eiωt1pxq is orbitally stable.
+Proof of Theorem 3.5.
+Assume that Lemma 3.6 holds and suppose, by contradiction, that uω
+is not orbitally stable. Hence, there exists 0 ă ε0 ă 2
+3η such that
+@n P N ∖ t0u, DuInit
+n
+P H1pTdq, }uInit
+n
+´ 1}H1 ă 1
+n and Dtn P r0, `8r, distpunptnq, O1q “ ε0,
+pt, xq ÞÑ unpt, xq P C0pr0, 8q; H1pTdqq being the solution of (13) with Cauchy data uInit
+n
+.
+To
+apply the coercivity estimate of Lemma 3.6, we define zn “
+´
+F p1q
+F punptnqq
+¯1{2
+unptnq. It is clear that
+zn P Sω since Fpznq “ Fp1q. Moreover,
+`
+unptnq
+˘
+nPN∖t0u is a bounded sequence in H1pTdq and
+limnÑ`8 Fpunptnqq “ Fp1q. Indeed, on the one hand, there exists γ P r0, 2πr such that
+}unptnq}H1 ď }unptnq ´ eiθ1}H1 ` }eiθ1}H1 ď 2dpunptnq, O1q ` }eiθ1}H1 “ 2ε0 ` }1}H1
+and, on the other hand,
+|Fpunptnqq ´ Fp1q| “ 1
+2|}unptnq}2
+L2 ´ }1}2
+L2| ď }unptnq ´ 1}L2pε0 ` }1}H1q ă 1
+npε0 ` }1}H1q.
+As a consequence, limnÑ`8 }zn ´ unptnq}H1 “ 0. This implies for n P N large enough,
+ε0
+2 ď dpzn, O1q ď 3ε0
+2
+ă η.
+Hence, thanks to Lemma 3.6, we obtain
+LωpuInit
+n
+q ´ Lωp1q “ Lωpunptnqq ´ Lωp1q “ Lωpunptnqq ´ Lωpznq ` Lωpznq ´ Lωp1q
+ě Lωpunptnqq ´ Lωpznq ` cdpzn, O1q2 ě Lωpunptnqq ´ Lωpznq ` c
+4ε2
+0.
+14
+
+Finally, using the fact that BuLωp1q “ 0 and B2
+uLωp1qpw, wq ď C}w}2
+H1, we deduce that
+lim
+nÑ`8pLωpuInit
+n
+q ´ Lωp1qq “ 0,
+lim
+nÑ`8pLωpunptnqq ´ Lωpznqq “ 0.
+We are thus led to a contradiction.
+Since BuLωp1q “ 0, the coercivity estimate (25) can be obtained from a similar estimate on the
+bilinear form B2
+uLωp1qpw, wq for any w P H1. As pointed out before, the difficulty lies in the fact
+that, in general, this bilinear form is not positive on the whole space H1. The following lemma
+states that it is enough to have a coercivity estimate on B2
+uLωp1qpw, wq for any w P T1SωXpT1O1qK.
+Recall that the tangent set to Sω is given by
+T1Sω “ tu P H1pTd; Cq, BuFp1q “ 0u “
+"
+pq, pq P H1pTd, Rq ˆ H1pTd, Rq,
+A ˆ
+q
+p
+˙ ˇˇˇ
+ˆ
+1
+0
+˙ E
+“ 0
+*
+This set is the orthogonal to 1 with respect to the inner product defined in (15). The tangent set
+to O1 (which is the orbit generated by the phase multiplication) is
+T1O1 “ spanRti1u
+so that
+pT1O1qK “ tu P H1pTd, Cq, xu, i1y “ 0u “
+"
+pq, pq : Td Ñ R,
+A ˆ
+q
+p
+˙ ˇˇˇ
+ˆ
+0
+1
+˙ E
+“ 0
+*
+.
+Lemma 3.7 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose that
+there exists ˜c ą 0
+B2
+uLωp1qpu, uq ě ˜c}u}2
+H1
+(26)
+for any u P T1S1 X pT1O1qK. Then there exist η ą 0 and c ą 0 such that (25) is satisfied.
+Proof.
+Let w P Sω such that distpw, O1q ă η with η ą 0 small enough. By means of an implicit
+function theorem argument (see [4, Section 9, Lemma 8]), we obtain that there exists θ P r0, 2πr
+and v P pT1O1qK such that
+eiθw “ 1 ` v,
+distpw, O1q ď }v}H1 ď Cdistpw, O1q
+for some positive constant C.
+Next, we use the fact that H1pTdq “ T1Sω ‘ spanRt1u to write v “ v1 ` v2 with v1 P T1Sω X
+pT1O1qK and v2 P spanRt1u X pT1O1qK. Since v “ eiθw ´ 1 and Fpwq “ Fp1q, we obtain
+0 “ Fpeiθwq ´ Fp1q “ 1
+2
+ˆ
+Td |v|2 dx ` Re
+ˆ
+Tdpv1 ` v2q1 dx “ 1
+2
+ˆ
+Td |v|2 dx ` Re
+ˆ
+Td v21 dx.
+Since v2 P spanRt1u, it follows that
+}v2}H1 ď }v}2
+H1
+}1}L2 .
+This implies
+}v1}H1 “ }v ´ v2}H1 ě }v}H1 ´
+1
+}1}L2 }v}2
+H1 ě 1
+2}v}H1
+15
+
+provided }v}H1 ď }1}L2. As a consequence, if }v}H1 is small enough, using that B2
+uLωp1qpw, zq ď
+C}w}H1}z}H1, we obtain
+B2
+uLωp1qpv1, v2q ď C}v}3
+H1,
+B2
+uLωp1qpv2, v2q ď C}v}4
+H1.
+This leads to
+B2
+uLωp1qpv, vq “ B2
+uLωp1qpv1, v1q ` op}v}2
+H1q.
+Finally, let w P Sω be such that dpw, O1q ă η. We have
+Lωpwq ´ Lωp1q “ Lωpeiθwq ´ Lωp1q “ 1
+2B2
+uLωp1qpv, vq ` op}v}2
+H1q
+“ 1
+2B2
+uLωp1qpv1, v1q ` op}v}2
+H1q ě ˜c}v1}2
+H1 ` op}v}2
+H1q ě ˜c
+2}v}2
+H1 ` op}v}2
+H1q
+ě ˜c
+4distpw, O1q2
+where we use BuLωp1q “ 0 and v1 P T1Sω X pT1O1qK.
+At the end of the day, to prove the orbital stability of the plane wave uωpt, xq “ eiωt1pxq it
+is enough to prove (26) for any u P T1S1 X pT1O1qK. This can be done by studying the spectral
+properties of the operator S. However, in the simpler case of the Hartree equation, the coercivity
+of B2
+uLωp1q on T1S1 X pT1O1qK can be also obtained directly from the expression
+B2
+uLωp1qpu, uq “ Re
+ˆˆ
+Td
+ˆ
+´∆u
+2 ´ 2γ2κΣ ‹ Repuq
+˙
+upxq dx
+˙
+“ xSu|uy.
+(27)
+Let u P T1S1 X pT1O1qK and write u “ q ` ip. This leads to
+B2
+uLωp1qpu, uq “ 1
+2
+ˆ
+Td |∇q|2 dx ´ 2γ2κ
+ˆ
+TdpΣ ‹ qqq dx ` 1
+2
+ˆ
+Td |∇p|2 dx.
+Moreover, since u P T1S1 X pT1O1qK, we have
+ˆ
+Td q dx “ 0 and
+ˆ
+Td p dx “ 0.
+As a consequence, thanks to the Poincaré-Wirtinger inequality, we deduce
+B2
+uLωp1qpu, uq ě 1
+2
+ˆ
+Td |∇q|2 dx ´ 2γ2κ
+ˆ
+TdpΣ ‹ qqq dx ` 1
+4}p}2
+H1.
+(28)
+Next, we expand q and Σ in Fourier series, i.e.
+qpxq “
+ÿ
+mPZd
+qmeim¨x and Σpxq “
+ÿ
+mPZd
+Σmeim¨x.
+Note that, if Σ “ σ1 ‹ σ1, then Σm “ p2πqdσ2
+1,m. Moreover,
+´
+Td q dx “ 0 implies q0 “ 0. Hence,
+1
+2
+ˆ
+Td |∇q|2 dx ´ 2γ2κ
+ˆ
+TdpΣ ‹ qqq dx “ p2πqd
+ÿ
+mPZd∖t0u
+ˆm2
+2 ´ 2γ2κp2πqdΣm
+˙
+q2
+m
+“ p2πqd
+ÿ
+mPZd∖t0u
+ˆ
+1 ´ 4γ2κp2πqd Σm
+m2
+˙ m2
+2 q2
+m.
+(29)
+As a consequence, we obtain the following statement.
+16
+
+Proposition 3.8 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose
+that there exists δ P p0, 1q such that
+4γ2κp2πq2d σ2
+1,m
+m2 ď δ
+(30)
+for all m P Zd ∖ t0u. Then, there exists ˜c ą 0 such that
+B2
+uLωp1qpu, uq ě ˜c}u}2
+H1
+(31)
+for any u P T1S1 X pT1O1qK.
+Proof.
+If (30) holds, then (28)-(29) lead to
+B2
+uLωp1qpu, uq ě 1 ´ δ
+2
+p2πqd
+ÿ
+mPZd∖t0u
+m2q2
+m ` 1
+4}p}H1 “ 1 ´ δ
+2
+}∇q}2
+L2 ` 1
+4}p}2
+H1 ě 1 ´ δ
+4
+}u}2
+H1.
+where in the last inequality we used the Poincaré-Wirtinger inequality together with the fact that
+´
+Td q dx “ 0.
+Remark 3.9 By decomposing the linear operator S into real and imaginary part and by using
+Fourier series, one can study its spectrum. In particular, S has exactly one negative eigenvalue
+λ´ “ ´2γ2κ
+@
+Σ
+D
+Td with eigenspace spanRt1u. Moreover, KerpSq “ spanRti1u. Finally, if (30) is
+satisifed, then infpσpSq X p0, 8qq ě 1´δ
+2 . Then, by applying the same arguments as in [5, Section
+6], we can recover the coercivity of B2
+uLωp1q on T1S1 X pT1O1qK.
+Finally, Proposition 3.8 together with Lemma 3.7 and Lemma 3.6, gives Theorem 3.5 and the
+orbital stability of the plane wave uω.
+4
+Stability analysis of the Schrödinger-Wave system:
+the case k “ 0
+Like in the case of the Hartree system, to study the stability of the plane wave solutions of the
+Schrödinger-Wave system (3a)-(3c), it is useful to write its solutions in the form
+Upt, xq “ eik¨xupt, xq
+with pt, x, zq ÞÑ pupt, xq, Ψpt, x, zqq solution to
+iBtu ` 1
+2∆xu ´ k2
+2 u ` ik ¨ ∇xu “ ´
+ˆ
+γσ1 ‹
+ˆ
+Rn σ2Ψ dz
+˙
+u,
+1
+c2 B2
+ttΨ ´ ∆zΨ “ ´γσ2σ1 ‹ |u|2.
+(32)
+If k P Zd and ω ą 0 satisfy the dispersion relation (12),
+uωpt, xq “ eiωt1pxq,
+Ψ˚pt, x, zq “ ´γΓpzq
+@
+σ1
+D
+Td,
+Π˚pt, x, zq “ BtΨ˚pt, x, zq “ 0
+17
+
+with Γ the solution of ´∆zΓ “ σ2 (see (11)), is a solution to (32) with initial condition
+uωp0, tq “ 1pxq,
+Ψ˚p0, x, zq “ ´γΓpzq
+@
+σ1
+D
+Td,
+Π˚p0, x, zq “ 0.
+For the time being, we stick to the framework identified for the study of the asymptotic Hartree
+equation. Problem (32) has a natural Hamiltonian symplectic structure when considered on the
+real Banach space H1pTdqˆH1pTdqˆL2pTd; .H1pRnqqˆL2pTd ˆRnq. Indeed, if we write u “ q `ip,
+with p, q real-valued, we obtain
+Bt
+¨
+˚
+˚
+˝
+q
+p
+Ψ
+Π
+˛
+‹‹‚“
+ˆJ
+0
+0
+´J
+˙
+∇pq,p,Ψ,ΠqHSW pq, p, Ψ, Πq
+with
+J “
+ˆ 0
+1
+´1
+0
+˙
+and
+HSWpq, p, Ψ, Πq “ 1
+2
+ˆ1
+2
+ˆ
+Td |∇q|2 ` |∇p|2 dx ` k2
+2
+ˆ
+Tdpp2 ` q2q dx ´
+ˆ
+Td pk ¨ ∇q dx `
+ˆ
+Td qk ¨ ∇p dx
+˙
+` 1
+4
+ˆ
+TdˆRn
+ˆΠ2
+c2 ` |∇zΨ|2
+˙
+dx dz
+` γ
+2
+ˆ
+Td
+ˆˆ
+TdˆRnpσ1px ´ yqσ2pzqΨpt, y, zq dy dz
+˙
+pp2 ` q2qpxq dx.
+Coming back to u “ q ` ip, we can write
+HSWpu, Ψ, Πq “ 1
+2
+ˆ1
+2
+ˆ
+Td |∇u|2 dx ` k2
+2
+ˆ
+Td |upxq|2 dx `
+ˆ
+Td k ¨ p´i∇uqu dx
+˙
+` 1
+4
+ˆ
+TdˆRn
+ˆΠ2
+c2 ` |∇zΨ|2
+˙
+dx dz
+` γ
+2
+ˆ
+Td
+ˆˆ
+TdˆRnpσ1px ´ yqσ2pzqΨpt, y, zq dy dz
+˙
+|upxq|2 dx.
+(33)
+As a consequence, HSW is a constant of the motion. Moreover, it is clear that (32) is invariant
+under multiplications by a phase factor of u so that Fpuq “ 1
+2}u}2
+L2 is conserved by the dynamics.
+However, now, the quantities
+Gjpuq “ 1
+2
+ˆ
+Td
+ˆ1
+i Bxju
+˙
+u dx
+(34)
+are not constants of the motion:
+d
+dtGjpuqptq “ γ
+2
+ˆ
+Td
+ˆ
+Td Bxjσ1px ´ yq
+ˆˆ
+Rn σ2pzqΨpt, y, zq dz
+˙
+|u|2pt, xq dy dx.
+As a consequence, they cannot be used in the construction of the Lyapounov functional as we did
+for the Hartree system (see (23)).
+18
+
+Finally, we consider the Banach space H1pTdqˆH1pTdqˆL2pTd; .H1pRnqqˆL2pTdˆRnq endowed
+with the inner product
+C
+¨
+˚
+˚
+˝
+q
+p
+Ψ
+Π
+˛
+‹‹‚
+ˇˇˇ
+¨
+˚
+˚
+˝
+q1
+p1
+Ψ1
+Π1
+˛
+‹‹‚
+G
+“
+ˆ
+Td
+`
+pp1 ` qq1q dx `
+ˆ
+TdˆRnp∇zΨ∇zΨ1 ` ΠΠ1q dx dz
+that can be also interpreted as an inner product for complex valued functions:
+xpu, Ψ, Πq|pu1, Ψ1, Π1qy “ Re
+ˆ
+Td uu1 dx `
+ˆ
+TdˆRnp∇zΨ ¨ ∇zΨ1 ` ΠΠ1q dx dz.
+(35)
+We denote by } ¨ } the norm on H1pTdq ˆ L2pTd; .H1pRnqq ˆ L2pTd ˆ Rnq induced by this inner
+product.
+4.1
+Preliminary results for the linearized problem: spectral sta-
+bility when k “ 0
+As before, we linearize the system (3a)-(3c) around the plane wave solution obtained in Section 2.1.
+Namely, we expand
+Upt, xq “ Ukpt, xqp1 ` upt, xqq,
+Ψpt, x, zq “ ´γ
+@
+σ1
+D
+TdΓpzq ` ψpt, x, zq
+and, assuming that u, ψ and their derivatives are small, we are led to the following equations for
+the fluctuation pt, xq ÞÑ upt, xq P C, pt, x, zq ÞÑ ψpt, x, zq P R
+iBtu ` 1
+2∆xu ` ik ¨ ∇xu “ γΦ,
+´ 1
+c2 B2
+ttψ ´ ∆zψ
+¯
+pt, x, zq “ ´γσ2pzqσ1 ‹ ρpt, xq,
+ρpt, xq “ 2Re
+`
+upt, xq
+˘
+,
+Φpt, xq “
+¨
+TdˆRn σ1px ´ yqσ2pzqψpt, y, zq dz dy.
+(36)
+We split the solution into real and imaginary parts
+upt, xq “ qpt, xq ` ippt, xq,
+qpt, xq “ Repupt, xqq,
+ppt, xq “ Impupt, xqq.
+We obtain
+pBtq ` 1
+2∆xp ` k ¨ ∇xqqpt, xq “ 0,
+pBtp ´ 1
+2∆xq ` k ¨ ∇xpqpt, xq “ ´γ
+ˆ
+σ1 ‹
+ˆ
+Rn σ2pzqψpt, ¨, zq dz
+˙
+pxq,
+´ 1
+c2 B2
+ttψ ´ ∆zψ
+¯
+pt, x, zq “ ´2γσ2pzqσ1 ‹ qpt, xq.
+(37)
+19
+
+It is convenient to set
+π “ ´ 1
+2c2 Btψ,
+in order to rewrite the wave equation as a first order system. We obtain
+Bt
+¨
+˚
+˚
+˝
+q
+p
+ψ
+π
+˛
+‹‹‚“ Lk
+¨
+˚
+˚
+˝
+q
+p
+ψ
+π
+˛
+‹‹‚
+(38)
+where Lk is the operator defined by
+Lk :
+¨
+˚
+˚
+˝
+q
+p
+ψ
+π
+˛
+‹‹‚ÞÝÑ
+¨
+˚
+˚
+˚
+˚
+˚
+˚
+˝
+´1
+2∆xp ´ k ¨ ∇xq
+1
+2∆xq ´ k ¨ ∇xp ´ γσ1 ‹
+ˆˆ
+Rn σ2ψ dz
+˙
+´2c2π
+´1
+2∆zψ ` γσ2σ1 ‹ q
+˛
+‹‹‹‹‹‹‚
+For the next step, we proceed via Fourier analysis as before. We expand q, p, ψ, π and σ1 by means
+of their Fourier series:
+ψpt, x, zq “
+ÿ
+mPZd
+ψmpt, zqeim¨x,
+ψmpt, zq “
+1
+p2πqd
+ˆ
+Td ψpt, x, zqe´im¨x dx,
+πpt, x, zq “
+ÿ
+mPZd
+πmpt, zqeim¨x,
+πmpt, zq “
+1
+p2πqd
+ˆ
+Td πpt, x, zqe´im¨x dx.
+Moreover, recall that σ1 being real and radially symmetric, (19) holds and, by definition,
+@
+σ1
+D
+Td “
+p2πqdσ1,0.
+As a consequence, since the Fourier modes are uncoupled, the Fourier coefficients
+pQmptq, Pmptq, ψmpt, zq, πmpt, zqq
+satisfy
+Bt
+¨
+˚
+˚
+˝
+Qm
+Pm
+ψm
+πm
+˛
+‹‹‚“ Lk,m
+¨
+˚
+˚
+˝
+Qm
+Pm
+ψm
+πm
+˛
+‹‹‚
+(39)
+where Lk,m stands for the operator defined by
+Lk,m
+¨
+˚
+˚
+˝
+Qm
+Pm
+ψm
+πm
+˛
+‹‹‚“
+¨
+˚
+˚
+˚
+˚
+˚
+˚
+˚
+˝
+´ik ¨ mQm ` m2
+2 Pm
+´m2
+2 Qm ´ ik ¨ mPm ´ γp2πqdσ1,m
+ˆ
+Rn σ2pzqψm dz
+´2c2πm
+γp2πqdσ2pzqσ1,mQm ´ 1
+2∆zψm
+˛
+‹‹‹‹‹‹‹‚
+.
+Like for the Hartree equation, the behavior of the mode m “ 0 can be analysed explicitly.
+20
+
+Lemma 4.1 (The mode m “ 0) For any k P Zd, the kernel of Lk,0 is spanned by p0, 1, 0, 0q.
+Moreover, equation (39) for m “ 0 admits solutions which grow linearly with time.
+Proof.
+Let pQ0, P0, ψ0, π0q P KerpLk,0q. It means that
+$
+’
+’
+’
+&
+’
+’
+’
+%
+γp2πqdσ1,0
+ˆ
+Rn σ2pzqψ0pzq dz “ 0,
+π0 “ 0,
+∆zψ0 “ 2γp2πqdσ2pzqσ1,0Q0,
+which yields ψ0pzq “ ´2γ
+@
+σ1
+D
+TdQ0Γpzq with Γpzq “ p´∆q´1σ2pzq so that
+´2γ2@
+σ1
+D2
+TdκQ0 “ 0.
+It implies that Q0 “ 0, ψ0 “ 0 while P0 is left undetermined.
+For m “ 0, the first equation in (39) tells us that Q0ptq “ Q0p0q P C is constant. Next, we get
+Btψ0 “ ´2c2π0 which leads to
+B2
+ttψ0 ´ c2∆zψ0 “ ´σ2pzq 2γc2@
+σ1
+D
+TdQ0p0q
+looooooooomooooooooon
+:“C1
+(40)
+The solution of (40) with initial condition pψ0pzq, π0pzq “ ´ 1
+2c2Btψp0, zqq P .H1pRnqˆL2pRnq satisfies
+pψ0pt, ξq “ pψ0p0, ξq cospc|ξ|tq ´ 2c2pπ0pξqsinpc|ξ|tq
+c|ξ|
+´
+ˆ t
+0
+sinpc|ξ|sq
+c|ξ|
+pσ2pξqC1 ds
+where pψ0pt, ξq and pπ0pt, ξq are the Fourier transforms of z ÞÑ ψpt, zq and z ÞÑ πpt, zq respectively.
+Finally, integrating
+BtP0 “ ´γ
+@
+σ1
+D
+Td
+loooomoooon
+:“C2
+ˆ
+Rn σ2pzqψ0pzq dz
+we obtain
+P0ptq “ P0p0q ` C2
+ˆ
+Rn pσ2pξq pψ0p0, ξqsinpc|ξ|tq
+c|ξ|
+dξ
+p2πqn ´ 2c2C2
+ˆ
+Rn pσ2pξqpπ0p0, ξq1 ´ cospc|ξ|tq
+c2|ξ|2
+dξ
+p2πqn
+´ C1C2
+ˆ t
+0
+ˆ s
+0
+pcpτq dτ ds
+where
+pcpτq “
+ˆ
+Rd |pσ2pξq|2 sinpc|ξ|τq
+c|ξ|
+dξ
+p2πqn .
+This kernel already appears in the analysis performed in [10, 19]. The contribution involving the
+initial data of the vibrational field can be uniformly bounded by
+1
+p2πqn
+ˆˆ
+Rd
+|pσ2pξq|2
+c2|ξ|2 dξ
+˙1{2 #ˆˆ
+Rd | pψ0p0, ξq|2 dξ
+˙1{2
+` 4c2
+ˆˆ
+Rd
+|pπ0p0, ξq|2
+c2|ξ|2
+dξ
+˙1{2+
+.
+Next, as a consequence of (H2), it turns out that pc is compactly supported, with
+´ 8
+0 pcpτq dτ “ κ
+c2,
+see [10, Lemma 14] and [19, Section 2.4]. It follows that
+ˆ t
+0
+ˆ s
+0
+pcpτq dτ ds “
+ˆ t
+0
+pcpτq
+ˆˆ t
+τ
+ds
+˙
+dτ “
+ˆ t
+0
+pt ´ τqpcpτq dτ
+„
+tÑ8 t κ
+c2 ´
+ˆ 8
+0
+τpcpτq dτ,
+21
+
+which concludes the proof.
+When k “ 0, basic estimates based on the energy conservation allow us to justify the stability
+of the solutions with zero mean. However, in contrast to what has been established for the Hartree
+system, this analysis does not extend to any mode k � 0, since the system is not Galilean invariant.
+Theorem 4.2 (Linearized stability for the Schrödinger-Wave system when k “ 0) Let k “
+0. Suppose (9) and let pu, ψ, πq be the solution of (36) associated to an initial data uInit P H1pTdq, ψInit P
+L2pTd; .H1pRnqq, πInit P L2pTd ˆ Rnq such that
+´
+Td uInit dx “ 0. Then, there exists a constant C ą 0
+such that suptě0 }upt, ¨q}H1 ď C.
+Proof.
+Again, we use the energetic properties of the linearized equation (36). We have already
+remarked that
+´
+Td upt, xq dx “ 0 for any t ě 0 when
+´
+Td uInit dx “ 0. We start by computing
+d
+dt
+"1
+2
+ˆ
+Td |∇xu|2 dx ` 1
+2
+ˆ
+TdˆRn
+´|Btψ|2
+c2
+` |∇zψ|2¯
+dz dx
+*
+“ ´iγ
+2
+ˆ
+Td Φ∆xpu ´ uq dx ´ γ
+ˆ
+TdˆRn Btψσ2σ1 ‹ pu ` uq dz dx.
+Next, we get
+d
+dt
+ˆ
+Td Φpu ` uq dx
+“
+ˆ
+TdˆRn Btψσ2σ1 ‹ pu ` uq dz dx
+` i
+2
+ˆ
+Td Φ∆xpu ´ uq dx ´
+ˆ
+Td Φk ¨ ∇xpu ` uq dx.
+We get rid of the last term by assuming k “ 0 and we arrive in this case at
+d
+dt
+"1
+2
+ˆ
+Td |∇xu|2 dx ` 1
+2
+ˆ
+TdˆRn
+´|Btψ|2
+c2
+` |∇zψ|2¯
+dz dx ` γ
+ˆ
+Td Φpu ` uq dx
+*
+“ 0.
+We estimate the coupling term as follows
+ˇˇˇˇ
+ˆ
+Td Φpu ` uq dx
+ˇˇˇˇ “
+ˇˇˇˇ
+ˆ
+TdˆRn σ2pzqψpt, x, zqσ1 ‹ pu ` uqpt, xq dz dx
+ˇˇˇˇ
+ď }σ1 ‹ pu ` uq}L2 ˆ
+ˆˆ
+Td
+ˇˇˇ
+ˆ
+Rn σ2pzqψpt, x, zq dz
+ˇˇˇ
+2
+dx
+˙1{2
+ď }σ1}L1}u ` u}L2 ˆ
+ˆˆ
+Td
+ˇˇˇ
+ˆ
+Rn pσ2pξq pψpt, x, ξq
+dξ
+p2πqn
+ˇˇˇ
+2
+dx
+˙1{2
+ď 2}σ1}L1}u}L2 ˆ
+ˆˆ
+Td
+ˇˇˇ
+ˆ
+Rn
+pσ2pξq
+|ξ| |ξ|| pψpt, x, ξq|
+dξ
+p2πqn
+ˇˇˇ
+2
+dx
+˙1{2
+ď 2}σ1}L1}u}L2 ˆ
+ˆˆ
+Rn
+|pσ2pξq|2
+|ξ|2
+dξ
+˙1{2
+ˆ
+ˆˆ
+TdˆRn |ξ|2| pψpt, x, ξq|2
+dξ
+p2πqn dx
+˙1{2
+ď 2 ?κ}σ1}L1}u}L2 ˆ
+ˆˆ
+TdˆRn |∇zψpt, x, ξq|2 dz dx
+˙1{2
+“ 2 ?κ}σ1}L1}u}L2}∇zψ}L2
+ď 1
+2γ }∇zψ}2
+L2 ` 2κγ}σ1}2
+L1}u}2
+L2.
+22
+
+By using the Poincaré-Wirtinger inequality }u}L2 ď }∇xu}L2, we deduce that
+1
+2
+ˆ
+Td |∇xupt, xq|2 dx ď
+E0
+1 ´ 4γ2κ}σ1}2
+L1
+,
+where E0 depends on the energy of the initial state.
+While it is natural to start with the linearized operator Lk in (38), it turns out that this
+formulation is not well-adapted to study the spectral stability issue. The difficulties relies on the
+fact that the wave part of the system induces an essential spectrum, reminiscent to the fact that
+σessp´∆zq “ r0, 8q. For instance, this is even an obstacle to set up a perturbation argument from
+the Hartree equation, in the spirit of [15]. We shall introduce later on a more adapted formulation
+of the linearized equation, which will allow us to overcome these difficulties (and also to go beyond
+a mere perturbation analysis).
+4.2
+Orbital stability for the Schrödinger-Wave system when k “ 0
+In this subsection, we wish to establish the orbital stability of the plane wave solution to (32)
+obtained in Section 2.1, namely
+uωpt, xq “ eiωt1pxq,
+Ψ˚pt, x, zq “ ´γΓpzq
+@
+σ1
+D
+Td,
+Π˚pt, x, zq “ 0
+with k P Zd and ω ą 0 that satisfy the dispersion relation (12) and Γpzq “ p´∆q´1σ2pzq. The
+system (32) being invariant under multiplications of u by a phase factor, we define the corresponding
+orbit through p1pxq, ´γΓpzq
+@
+σ1
+D
+Td, 0q by
+O1 “ tpeiθ, ´γΓpzq
+@
+σ1
+D
+Td, 0q, θ P Ru.
+As before, orbital stability intuitively means that the solutions of (32) associated to initial data
+close enough to p1pxq, ´γΓpzq
+@
+σ1
+D
+Td, 0q remain at a close distance to the set O1.
+Let us introduce, for any k P Zd and ω ą 0 satisfying the dispersion relation (12), the set
+Sω “
+!
+pu, Ψ, Πq P H1pTd; Cq ˆ L2pTd; .H1pRnqq ˆ L2pTd, L2pRnqq, Fpuq “ Fp1q “ p2πqd
+2
+)
+,
+and the functional
+Lω,kpu, Ψ, Πq “ HSWpu, Ψ, Πq ` ωFpuq,
+(41)
+intended to serve as a Lyapounov functional, where HSW is the constant of motion defined in (33).
+For further purposes, we simply denote Lω “ Lω,0. Note that
+Lω,kpu, Ψ, Πq “ HSWpu, Ψ, Πq ` 1
+2i
+ˆ
+Td k ¨ ∇u ¯u dx
+loooooooooomoooooooooon
+“
+dÿ
+j“1
+kjGjpuq
+`
+´
+ω ` k2
+2
+¯
+Fpuq
+with HSW defined in (8) and Gjpuq defined in (34). Thanks to the dispersion relation (12), only the
+second term of this expression depends on k. Unfortunately, as pointed out before, the quantities
+23
+
+Gjpuq are not constants of the motion so that the dependence on k of the Lyapounov functional
+(41) cannot be disregarded, in contrast to what we did for the Hartree system in (23).
+Next, as in subsection 3.2, we need to evaluate the first and second order variations of Lω,k.
+We compute
+Bpu,Ψ,ΠqHSWpu, Ψ, Πqpv, φ, τq
+“ Re
+ˆ1
+2
+ˆ
+Tdp´∆uqv dx ` γ
+ˆ
+Td
+ˆ¨
+TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dz dy
+˙
+upxqvpxq dx
+˙
+` γ
+2
+ˆ
+Td
+ˆ¨
+TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy
+˙
+|upxq|2 dy dx
+` 1
+2
+¨
+TdˆRn
+´ 1
+c2 Π τ ` p´∆zΨq φ dz
+¯
+dx
+and
+B2
+pu,Ψ,ΠqHSWpu, Ψ, Πq
+`
+pv, φ, τq, pv1, φ1, τ 1q
+˘
+“ Re
+"1
+2
+ˆ
+Tdp´∆vqv1 dx
+`γ
+ˆ
+Td
+ˆ¨
+TdˆRn σ1px ´ yqσ2pzqpφpt, y, zqv1pxq ` φ1pt, y, zqvpxqq dz dy
+˙
+upxq dx
+˙
+`γ
+ˆ
+Td
+ˆ¨
+TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dz dy
+˙
+vpxqv1pxq dx
+˙*
+` 1
+2
+¨
+TdˆRn
+´ 1
+c2 τ τ 1 ` p´∆zφq φ1 dz
+¯
+dx.
+Besides, we have
+BuFpuqpvq “ Re
+ˆˆ
+Td uv dx
+˙
+,
+B2
+uFpuqpv, v1q “ Re
+ˆˆ
+Td vv1 dx
+˙
+,
+BuGjpuqpvq “ Im
+`´
+Td Bxjuv dx
+˘
+,
+B2
+uGpuqpv, v1q “ Im
+`´
+Td Bxjv1v dx
+˘
+.
+Accordingly, we are led to
+Bpu,Ψ,ΠqLω,kp1, ´γ
+@
+σ1
+D
+TdΓ, 0qpv, φ, τq
+“ Re
+ˆ
+´γ2@
+σ1
+D2
+Tdκ
+ˆ
+Td v dx `
+´
+ω ` k2
+2
+¯ ˆ
+Td v dx ` γ
+2
+@
+σ1
+D
+Td
+¨
+TdˆRnpσ2 ` ∆zΓq φ dz dx
+˙
+“ 0
+thanks to the dispersion relation (12) and the definition of Γ. Similarly, the second order derivative
+24
+
+casts as
+B2
+pu,Ψ,ΠqLω,kp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pv, φ, τq, pv, φ, τq
+˘
+“ Re
+ˆ1
+2
+ˆ
+Tdp´∆vqv dx ` 1
+2
+¨
+TdˆRn
+´τ 2
+c2 ` p´∆zφq φ dz
+¯
+dx
+` 2γ
+ˆ
+Td
+ˆ¨
+TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy
+˙
+vpxq dx
+´ γ2@
+σ1
+D
+Td
+ˆ
+Td
+ˆ¨
+TdˆRn σ1px ´ yqσ2pzqΓpzq dz dy
+˙
+vpxqvpxq dx `
+´
+ω ` k2
+2
+¯ ˆ
+Td vpxqvpxq dx
+˙
+` Im
+˜ dÿ
+j“1
+kj
+ˆ
+Td Bxjvv dx
+¸
+.
+The forth and fifth integrals combine as
+ˆ
+Td
+´
+ω ` k2
+2 ´ γ2κ
+@
+σ1
+D2
+Td
+¯
+vpxqvpxq dx “ 0
+which cancels out by virtue of the dispersion relation (12). Hence we get
+B2
+pu,Ψ,ΠqLω,kp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pv, φ, τq, pv, φ, τq
+˘
+“ Re
+ˆ1
+2
+ˆ
+Tdp´∆vqv dx ` 1
+2
+¨
+TdˆRn
+´τ 2
+c2 ` p´∆zφq φ dz
+¯
+dx
+` 2γ
+ˆ
+Td
+ˆ¨
+TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy
+˙
+vpxq dx ´ i
+ˆ
+Td k ¨ ∇v v dx
+˙
+.
+Remark 4.3 Note that the following continuity estimate holds: for any pv, φ, τq P H1pTd; Cq ˆ
+L2pTd; .H1pRnqq ˆ L2pTd ˆ Rnq,
+B2
+pu,Ψ,ΠqLω,kp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pv, φ, τq, pv, φ, τq
+˘
+ď 1
+2}∇v}2
+L2 ` 1
+2c2 }τ}2
+L2 ` 1
+2}φ}2
+L2x .
+H1z
+` 2γκ1{2}σ1}L1}v}L2}φ}L2x .
+H1z ` |k|}∇v}L2}v}L2 ď 1
+2
+ˆ
+p1 ` |k|q}v}2
+H1 ` 1
+c2 }τ}2
+L2 ` C}φ}2
+L2x .
+H1z
+˙
+ď maxp1{c2, 1 ` |k|, Cq
+2
+}pv, φ, τq}2
+with C “ 1 ` 4γ2κ}σ1}2
+L1.
+The functional Lω,k is conserved by the solutions of (32); however the difficulty relies on
+justifying its coercivity. We are only able to answer positively in the specific case k “ 0. Hence,
+the main result of this subsection restricts to this situation.
+Theorem 4.4 (Orbital stability for the Schrödinger-Wave system) Let k “ 0 and ω ą 0
+such that the dispersion relation (12) is satisfied. Suppose (9) holds. Then the plane wave solution
+peiωt1pxq, ´γΓpzq
+@
+σ
+D
+Td, 0q is orbitally stable, i.e.
+@ε ą 0, Dδ ą 0, @pvInit, φInit, τ Initq P H1pTd; Cq ˆ L2pTd; .H1pRnqq ˆ L2pTd ˆ Rnq,
+}vInit ´ 1}H1 ` }φInit ` γΓ
+@
+σ
+D
+Td}L2x .
+H1z ` }τ Init}L2 ă δ ñ sup
+tě0
+distppvptq, φptq, τptqq, O1q ă ε
+(42)
+25
+
+where distppv, φ, τq, O1q “ infθPr0,2πr }v ´ eiθ1}H1 ` }φ ` γΓ
+@
+σ
+D
+Td}L2x .
+H1z ` }τ}L2 and pt, x, zq ÞÑ
+pvpt, xq, φpt, x, zq, τpt, x, zqq stands for the solution of (32) with Cauchy data pvInit, φInit, τ Initq.
+Using the same argument as in the case of Theorem 3.5, we can reduce the proof of Theorem
+4.4 to the following coercivity estimate on the Lyapounov functional (and this is where we use that
+Lω,k is a conserved quantity).
+Lemma 4.5 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose that
+there exist η ą 0 and c ą 0 such that @pw, ψ, χq P Sω,
+distppw, ψ, χq, O1q ă η ñ Lω,kppw, ψ, χqq ´ Lω,kpp1pxq, ´γΓpzq
+@
+σ
+D
+Td, 0qq ě cdistppw, ψ, χq, O1q2.
+(43)
+Then the the plane wave solution peiωt1pxq, ´γΓpzq
+@
+σ
+D
+Td, 0q is orbitally stable.
+As we have seen before, since Bpu,ψ,ΠqLω,kpp1, ´γΓpzq
+@
+σ
+D
+Td, 0qq “ 0, the coercivity estimate
+(43) can be obtained from an estimate on the bilinear form
+B2
+pu,ψ,ΠqLω,kpp1, ´γ
+@
+σ1
+D
+TdΓ, 0qqppu, φ, τq, pu, φ, τqq
+for any pu, φ, τq P T1Sω X pT1O1qK. Here the tangent set to Sω is given by
+T1Sω “
+"
+u P H1pTd; Cq, Re
+ˆˆ
+Td upxq1pxq dx
+˙
+“ 0
+*
+ˆ L2pTd; .H1pRnqq ˆ L2pTd ˆ Rnq.
+This set is the orthogonal to p1, 0, 0q with respect to the inner product defined in (35). The tangent
+set to O1 (which is the orbit generated by the phase multiplications of 1) is
+T1O1 “ spanRtpi1, 0, 0qu
+so that
+pT1O1qK “
+"
+u P H1pTd; Cq, Re
+ˆ
+i
+ˆ
+Td upxq1pxq dx
+˙
+“ 0
+*
+ˆ L2pTd; .H1pRnqq ˆ L2pTd ˆ Rnq.
+Lemma 4.6 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose that
+there exists ˜c ą 0
+B2
+pu,ψ,ΠqLω,kpp1, ´γΓpzq
+@
+σ
+D
+Td, 0qqppu, φ, τq, pu, φ, τqq ě ˜cp}u}2
+H1 ` }φ}2
+L2x .
+H1z ` }τ}2
+L2q “ ˜c}pu, φ, τq}2
+(44)
+for any pu, φ, τq P T1S1 X pT1O1qK. Then there exist η ą 0 and c ą 0 such that (43) is satisfied.
+Proof.
+Let pw, ψ, χq P Sω such that distppw, ψ, χq, O1q ă η with η ą 0 small enough. Hence,
+infθPr0,2πq }w´eiθ1} ă η and, by means of an implicit function theorem argument (see [4, Section 9,
+Lemma 8]), we obtain that there exists θ P r0, 2πq and v P
+␣
+u P H1pTd; Cq, Re
+`
+i
+´
+Td upxq dx
+˘
+“ 0
+(
+such that
+eiθw “ 1 ` v,
+inf
+θPr0,2πq }w ´ eiθ1} ď }v}H1 ď C
+inf
+θPr0,2πq }w ´ eiθ1}
+26
+
+for some positive constant C. Denote by φpx, zq “ ψpx, zq`γΓpzq
+@
+σ1
+D
+Td. Then pv, φ, χq P pT1O1qK
+and }pv, φ, χq} ď Cη.
+Next, we use the fact that H1pTdq “
+␣
+u P H1pTd; Cq, Re
+`´
+Td upxq dx
+˘
+“ 0
+(
+‘ spanRt1u to write
+pv, φ, χq “ pv1, φ, χq ` pv2, 0, 0q with pv1, φ, χq P T1Sω X pT1O1qK and v2 P spanRt1u. Moreover,
+}v2}H1 ď }v}2
+H1
+}1}L2
+and
+}v1}H1 ě 1
+2}v}H1
+provided }v}H1 ď }1}L2. As a consequence, if }v}H1 is small enough, using that
+B2
+pu,Ψ,ΠqLω,kp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pv, φ, τq, pv1, φ1, τ 1q
+˘
+ď C}pv, φ, τq}}pv1, φ1, τ 1q},
+we obtain
+B2
+pu,Ψ,ΠqLω,kp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pv1, φ, χq, pv2, 0, 0q
+˘
+ď C}pv, φ, χq} }v}2
+H1 ď C}pv, φ, χq}3,
+B2
+pu,Ψ,ΠqLω,kp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pv2, 0, 0q, pv2, 0, 0q
+˘
+ď C}v}4
+H1 ď C}pv, φ, χq}4.
+This leads to
+B2
+pu,Ψ,ΠqLω,kp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pv, φ, χq, pv, φ, χq
+˘
+“ B2
+pu,Ψ,ΠqLω,kp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pv1, φ, χq, pv1, φ, χq
+˘
+` op}pv, φ, χq}2q.
+Finally, let pw, ψ, χq P Sω such that dppw, ψ, χq, O1q ă η, we have
+Lω,kppw, ψ, χqq ´ Lω,kpp1pxq, ´γΓpzq
+@
+σ
+D
+Td, 0qq “ Lω,kppeiθw, ψ, χqq ´ Lω,kpp1pxq, ´γΓpzq
+@
+σ
+D
+Td, 0qq
+“ B2
+pu,Ψ,ΠqLω,kp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pv, φ, χq, pv, φ, χq
+˘
+` op}pv, φ, χq}2q
+“ B2
+pu,Ψ,ΠqLω,kp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pv1, φ, χq, pv1, φ, χq
+˘
+` op}pv, φ, χq}2q
+ě ˜c}pv1, φ, τq}2 ` op}pv, φ, χq}2q ě ˜c
+2}pv, φ, τq}2 ` op}pv, φ, χq}2q
+ě ˜c
+6dppw, ψ, χq, O1q2
+where we use Bpu,Ψ,ΠqLω,kp1, ´γ
+@
+σ1
+D
+TdΓ, 0q “ 0 and pv1, φ, χq P T1Sω X pT1O1qK.
+As before, to prove the orbital stability of the plane solution peiωt1pxq, ´γΓpzq
+@
+σ
+D
+Td, 0q it is
+enough to prove (44) for any pu, φ, τq P T1S1 XpT1O1qK. Let pu, φ, τq P T1S1 XpT1O1qK and write
+u “ q ` ip with q, p P H1pTd; Rq. Then
+B2
+pu,Ψ,ΠqLω,kp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pu, φ, τq, pu, φ, τq
+˘
+“ Re
+ˆ1
+2
+ˆ
+Tdp´∆uqu dx ` 1
+2
+¨
+TdˆRn
+´τ 2
+c2 ` p´∆zφq φ dz
+¯
+dx
+` 2γ
+ˆ
+Td
+ˆ¨
+TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy
+˙
+upxq dx ´ i
+ˆ
+Td k ¨ ∇u u dx
+˙
+(45)
+can be reinterpreted as a quadratic form acting on the 4-uplet W “ pq, p, φ, τq. To be specific, it
+27
+
+expresses as the following quadratic form on W,
+QpW, Wq “1
+2
+ˆ
+Td |∇p|2 dx ` 1
+2c2
+¨
+TdˆRn |τ|2 dz dx ` 1
+2
+ˆ
+Td |∇q|2 dx ` 1
+2
+¨
+TdˆRnp´∆zφq φ dx dz
+` 2γ
+ˆ
+Td
+ˆ¨
+TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dyqpxq dx
+˙
+` 2
+ˆ
+Td qk ¨ ∇p dx.
+The crossed term
+´
+Td qk ¨ ∇p dx is an obstacle for proving a coercivity on Q.
+For this reason, let us focus on the case k “ 0. Since pu, φ, τq P T1S1 X pT1O1qK, we have
+ˆ
+Td q dx “ 0 and
+ˆ
+Td p dx “ 0.
+As a consequence, thanks to the Poincaré-Wirtinger inequality, we deduce, when k “ 0
+QpW, Wq ě1
+4}p}2
+H1 ` 1
+2c2 }τ}2
+L2 ` 1
+2
+ˆ
+Td |∇q|2 dx ` 1
+2
+¨
+TdˆRnp´∆zφq φ dx dz
+` 2γ
+ˆ
+Td
+ˆ¨
+TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy
+˙
+qpxq dx
+(46)
+Next, we expand q, σ1 and φp¨, zq in Fourier series, i.e.
+qpxq “
+ÿ
+mPZd
+qmeim¨x, φpx, zq “
+ÿ
+mPZd
+φmpzqeim¨x and σ1pxq “
+ÿ
+mPZd
+σ1,meim¨x.
+Note that σ1,m “ σ1,m “ σ1,´m since σ1 is real and radially symmetric. Moreover,
+´
+Td q dx “ 0
+implies q0 “ 0. Hence,
+ˆ
+Td
+ˆˆ
+TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy
+˙
+qpxq dx
+“ p2πq2dRe
+¨
+˝
+ÿ
+mPZd∖t0u
+σ1,mqm
+ˆ
+Rn σ2pzqφmpzq dz
+˛
+‚
+which implies
+1
+2
+ˆ
+Td |∇q|2 dx ` 1
+2
+¨
+TdˆRnp´∆zφq φ dx dz
+` 2γ
+ˆ
+Td
+ˆ¨
+TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy
+˙
+qpxq dx
+“ p2πqd
+ÿ
+mPZd∖t0u
+Re
+ˆm2
+2 q2
+m ` 1
+2
+ˆ
+Rn |∇zφm|2 dz ` 2p2πqdγσ1,mqm
+ˆ
+Rn σ2pzqφmpzq dz
+˙
+.
+Next, we remark that for any m P Zd,
+ˇˇˇˇRe
+ˆ
+2p2πqdγσ1,mqm
+ˆ
+Rn σ2pzqφmpzq dz
+˙ˇˇˇˇ ď 2p2πqdγσ1,m|qm| ?κ}∇φm}L2
+ď 1
+2˜δp4γ2κp2πq2dσ2
+1,mqq2
+m `
+˜δ
+2}∇φm}2
+L2
+28
+
+for any ˜δ ą 0. Finally, for any ˜δ P p0, 1q, we get
+1
+2
+ˆ
+Td |∇q|2 dx ` 1
+2
+¨
+TdˆRnp´∆zφq φ dx dz
+` 2γ
+ˆ
+Td
+ˆ¨
+TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy
+˙
+qpxq dx
+ě p2πqd ÿ
+mPZd
+ˆˆm2
+2 ´ 1
+2˜δ p4γ2κp2πq2dσ2
+1,mq
+˙
+q2
+m ` 1 ´ ˜δ
+2
+}∇φm}2
+L2
+˙
+(47)
+As a consequence, we obtain the following statement.
+Proposition 4.7 Let k “ 0 and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose
+that there exists δ P p0, 1q such that
+4γ2κp2πq2d σ2
+1,m
+m2 ď δ
+(48)
+for all m P Zd ∖ t0u. Then, there exists ˜c ą 0 such that
+B2
+pu,Ψ,ΠqLωp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pu, φ, τq, pu, φ, τq
+˘
+ě ˜c}pu, φ, τq}2
+(49)
+for any pu, φ, τq P T1S1 X pT1O1qK.
+Proof.
+If (48) holds, then, for any ˜δ P pδ, 1q, (45)-(46)-(47) lead to
+B2
+pu,Ψ,ΠqLωp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pu, φ, τq, pu, φ, τq
+˘
+ě 1
+4}p}2
+H1 ` 1
+2c2 }τ}2
+L2
+`
+˜δ ´ δ
+2˜δ p2πqd
+ÿ
+mPZd∖t0u
+m2q2
+m ` 1 ´ ˜δ
+2
+p2πqd ÿ
+mPZd
+}∇φm}2
+L2
+“ 1
+4}p}2
+H1 ` 1
+2c2 }τ}2
+L2 `
+˜δ ´ δ
+2˜δ }∇q}L2 ` 1 ´ ˜δ
+2
+}φ}L2xH1z ě ˜c}pu, φ, τq}2
+where in the last inequality we used the Poincaré-Wirtinger inequality together with the fact that
+´
+Td q dx “ 0.
+Finally, Proposition 4.7 together with Lemma 4.6 and Lemma 4.5 gives Theorem 4.4 and the
+orbital stability of the plane wave solution peiωt1pxq, ´γΓpzq
+@
+σ
+D
+Td, 0q in the case k “ 0.
+Remark 4.8 The coercivity of B2
+pu,Ψ,ΠqLωp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pu, φ, τq, pu, φ, τq
+˘
+on T1S1XpT1O1qK
+can be recovered from the spectral properties of a convenient unbounded linear operator S. Indeed,
+as we have seen before, by decomposing u into real and imaginary part, the quadratic form defined
+by (45) (with k “ 0) can be written as
+QpW, Wq “ 1
+2
+ˆ
+Td |∇p|2 dx ` 1
+2c2
+¨
+TdˆRn |τ|2 dz dx `
+B
+S
+ˆq
+φ
+˙ ˇˇˇ
+ˆq
+φ
+˙F
+29
+
+with S : H2pTdq ˆ L2pTd; .H1pRnqq Ă L2pTdq ˆ L2pTd; .H1pRnqq Ñ L2pTdq ˆ L2pTd; .H1pRnqq the
+unbounded linear operator given by
+S
+ˆ
+q
+φ
+˙
+“
+¨
+˚
+˝
+´1
+2∆xq ` γσ1 ‹
+ˆ
+Rn σ2φ dz
+1
+2φ ` γΓσ1 ‹ q
+˛
+‹‚
+(where we remind the reader that Γ “ p´∆q´1σ2q) and the inner product
+Bˆ
+q
+φ
+˙ ˇˇˇ
+ˆ
+q1
+φ1
+˙F
+“
+ˆ
+Td qq1 dx`
+ˆ
+TdˆRn ∇zφ¨∇zφ1 dz dx “
+ˆ
+Td qq1 dx`
+ˆ
+TdˆRn
+ˆφpx, ξqˆφ1px, ξq|ξ|2 dξ
+p2πqn dx.
+Note that L2pTdq ˆ L2pTd; .H1pRnqq is an Hilbert space with this inner product since n ě 3.
+Since
+ˆ
+T d
+ˆ
+σ1 ‹
+ˆ
+Rn σ2φ dz
+˙
+pxqq1pxq dx “
+ˆ
+Td
+ˆˆ
+TdˆRn σ1px ´ yqσ2pzqφpy, zq dz dy
+˙
+q1pxq dx
+“
+ˆ
+TdˆRn φpx, zqσ2pzqpσ1 ‹ q1qpxq dx dz “
+ˆ
+TdˆRn
+ˆφpx, ξq ˆσ2pξq
+|ξ|2 pσ1 ‹ q1qpxq dx|ξ|2 dξ
+p2πqn
+we can check that S is a self-adjoint operator on L2pTdq ˆ L2pTd; .H1pRnqq. In particular, σpSq Ă R
+and one can easily study the spectrum of S.
+More precisely, using Fourier series, we find that if λ is an eigenvalue of S then there exists at
+least one m P Zd such that for some pqm, φmq � p0, 0q there holds
+$
+’
+’
+&
+’
+’
+%
+ˆm2
+2 ´ λ
+˙
+qm ` γp2πqdσ1,m
+ˆ
+Rn σ2pzqφmpzq dz “ 0,
+ˆ1
+2 ´ λ
+˙
+φmpzq ` γp2πqdΓpzqσ1,mqm “ 0.
+Let λ � 1
+2. Hence, for any m P Zd, qm “ 0 implies φmpzq “ 0 for any z P Rn. As a consequence,
+we may assume qm � 0. This leads to φmpzq “ ´ γp2πqdσ1,mqm
+1{2´λ
+Γpzq and
+ˆm2
+2 ´ λ
+˙ ˆ1
+2 ´ λ
+˙
+´ γ2p2πq2dσ2
+1,mκ “ 0.
+By solving this equation, we obtain
+λ˘,m “
+´
+m2`1
+2
+¯
+˘
+c´
+m2´1
+2
+¯2
+` 4γ2p2πq2dσ2
+1,mκ
+2
+so that λ`,m ě 1
+4 for any m P Zd. Next, we remark that
+λ´,0 “
+1
+2 ´
+b
+1
+4 ` 4γ2p2πq2dσ2
+1,0κ
+2
+ă 0
+30
+
+since 4γ2κp2πq2dσ2
+1,0 ą 0. This eigenvalue corresponds to an eigenfunction p˜q, ˜φq with ˜q P spanRt1u.
+In particular,
+´
+Td ˜qpxq dx � 0. Finally, if (30) holds,
+λ´,m ě
+´
+m2`1
+2
+¯
+´
+c´
+m2´1
+2
+¯2
+` δm2
+2
+ě 1 ´ δ
+5
+for any m P Zd ∖ t0u.
+We conclude that
+B
+S
+ˆq
+φ
+˙ ˇˇˇ
+ˆq
+φ
+˙F
+“
+C¨
+˚
+˝
+´1
+2∆xq ` γσ1 ‹
+ˆ
+Rn σ2φ dz
+1
+2φ ` γΓσ1 ‹ q
+˛
+‹‚
+ˇˇˇ
+ˆq
+φ
+˙G
+ě min
+ˆ1
+2, 1 ´ δ
+5
+˙
+p}q}2
+L2 ` }φ}L2x .
+H1z q
+for all pq, φq P tq P L2pTdq,
+´
+T d q dx “ 0u ˆ L2pTd; .H1pRnqq. This, together with the Poincaré-
+Wirtinger inequality, proves the coercivity of B2
+pu,Ψ,ΠqLωp1, ´γ
+@
+σ1
+D
+TdΓ, 0q
+`
+pu, φ, τq, pu, φ, τq
+˘
+on
+T1S1 X pT1O1qK.
+5
+Discussion about the case k � 0
+5.1
+A new symplectic form of the linearized Schrödinger-Wave
+system
+We go back to the linearized problem. The viewpoint presented in Section 4.1 looks quite natural;
+however, it misses some structural properties of the problem. In order to work in a unified functional
+framework, we find convenient to change the wave unknown ψ, which is naturally valued in .H1pRnq,
+into p´∆q´1{2φ, where the new unknown φ now lies in L2pRnq. Hence, the linearized problem is
+rephrased as
+BtX “ LX,
+where X stands for the 4-uplet pq, p, φ, πq and
+LX “
+¨
+˚
+˚
+˚
+˚
+˚
+˚
+˚
+˝
+´1
+2∆xp ´ k ¨ ∇xq
+1
+2∆xq ´ k ¨ ∇xp ´ γσ1 ‹
+ˆˆ
+Rnp´∆q´1{2σ2φ dz
+˙
+´2c2p´∆q1{2π
+1
+2p´∆q1{2φ ` γσ2σ1 ‹ q
+˛
+‹‹‹‹‹‹‹‚
+.
+The operator L is seen as an operator on the Hilbert space
+V “ L2pTdq ˆ L2pTdq ˆ L2pTd; L2pRnqq ˆ L2pTd; L2pRnqq,
+with domain DpLq “ H2pTdq ˆ H2pTdq ˆ L2pTd; H1pRnqq ˆ L2pTd; H1pRnqq. We can start with
+the following basic information, which has the consequence that the spectral stability amounts to
+justify that σpLq Ă iR.
+31
+
+Lemma 5.1 Let pλ, Xq be an eigenpair of L. Let Y : px, zq ÞÑ pqp´xq, ´pp´xq, φp´x, zq, ´πp´x, zqq.
+Then, pλ, Xq, p´λ, Y q and p´λ, Y q are equally eigenpairs of L.
+Proof.
+Since L has real coefficients, LX “ λX implies LX “ λX. Next, we check that
+LY px, zq
+“
+¨
+˚
+˚
+˚
+˚
+˚
+˚
+˚
+˝
+1
+2∆xp ` k ¨ ∇xq
+1
+2∆xq ´ k ¨ ∇xp ´ γσ1 ‹
+ˆˆ
+Rnp´∆q´1{2σ2φ dz1
+˙
+2c2p´∆q1{2π
+1
+2p´∆q1{2φ ` γσ2σ1 ‹ q
+˛
+‹‹‹‹‹‹‹‚
+p´x, zq
+“
+λ
+¨
+˚
+˚
+˝
+´qp´x, zq
+pp´x, zq
+´φp´x, zq
+πp´x, zq
+˛
+‹‹‚“ ´λY px, zq.
+Next, we make a new symplectic structure appear. To this end, let us introduce the blockwise
+operator
+J “
+ˆJ1
+0
+0
+J2
+˙
+,
+J1 “
+ˆ 0
+1
+´1
+0
+˙
+,
+J2 “
+ˆ
+0
+´p´∆q1{2
+p´∆q1{2
+0
+˙
+.
+We are thus led to
+L “ J L
+with
+L X “
+¨
+˚
+˚
+˚
+˚
+˚
+˚
+˝
+´1
+2∆xq ` k ¨ ∇xp ` γσ1 ‹
+ˆˆ
+Rnp´∆q´1{2σ2φ dz
+˙
+´1
+2∆xp ´ k ¨ ∇xq
+1
+2φ ` γp´∆q´1{2σ2σ1 ‹ q
+2c2π
+˛
+‹‹‹‹‹‹‚
+.
+(50)
+For further purposes, we also set
+Ă
+J “
+ˆ ˜
+J1
+0
+0
+˜
+J2
+˙
+,
+˜
+J1 “
+ˆ0
+´1
+1
+0
+˙
+,
+˜
+J2 “
+ˆ
+0
+p´∆q´1{2
+´p´∆q´1{2
+0
+˙
+.
+(51)
+The operator J has 0 in its essential spectrum; nevertheless
+Ă
+J plays the role of its inverse since
+J Ă
+J “ I “
+Ă
+J J .
+Lemma 5.2 The operator L is an unbounded self adjoint operator on V with domain DpL q “
+H2pTdq ˆ H2pTdq ˆ L2pTd; L2pRnqq ˆ L2pTd; L2pRnqq, and the operator J is skew-symmetric.
+Proof.
+The space V is endowed with the standard L2 inner product
+`
+X|X1q “
+ˆ
+Tdpqq1 ` pp1q dx `
+¨
+TdˆRnpφφ1 ` ππ1q dx dz.
+32
+
+We get
+`
+L X|X1˘
+“
+ˆ
+Td
+!´
+´ 1
+2∆xq ` k ¨ ∇xp
+¯
+q1 `
+´
+´ 1
+2∆xp ´ k ¨ ∇xq
+¯
+p1
+)
+dx
+`γ
+ˆ
+Td σ1 ‹
+ˆˆ
+Rnp´∆q´1{2σ2φ dz
+˙
+q1 dx
+`
+¨
+TdˆRn
+´1
+2φφ1 ` 2c2ππ1
+¯
+dx dz
+`γ
+¨
+TdˆRn
+´
+p´∆q´1{2σ2σ1 ‹ q
+¯
+φ1 dx dz
+“
+ˆ
+Td
+!
+q
+´
+´ 1
+2∆xq1 ` k ¨ ∇xp1
+¯
+` p
+´
+´ 1
+2∆xp1 ´ k ¨ ∇xq1
+¯)
+dx
+`γ
+¨
+TdˆRn φp´∆q´1{2σ2σ1 ‹ q1 dz dx
+`
+¨
+TdˆRn
+´1
+2φφ1 ` 2c2ππ1
+¯
+dx dz
+`γ
+ˆ
+Td qσ1 ‹
+ˆˆ
+Rnp´∆q´1{2σ2φ1 dz
+˙
+dx
+“
+`
+X|L X1˘
+,
+and
+`
+J X|X1˘
+“
+¨
+Td
+´
+pq1 ´ qp1
+¯
+dx `
+¨
+TdˆRn
+´
+´ p´∆q1{2πφ1 ` p´∆q1{2φπ1
+¯
+dx dz
+“
+´
+¨
+Td
+´
+qp1 ´ pq1
+¯
+dx ´
+¨
+TdˆRn
+´
+´ φp´∆q1{2π1 ` πp´∆q1{2φ1
+¯
+dx dz
+“
+´
+`
+X|J X1˘
+As said above, justifying the spectral stability for the Schrödinger-Wave equation reduces to
+verify that the spectrum σpLq is purely imaginary. However, the coupling with the wave equation
+induces delicate subtleties and a direct approach is not obvious. Instead, based on the expression
+L “ J L , we can take advantage of stronger structural properties. In particular, the functional
+framework adopted here allows us to overcome the difficulties related to the essential spectrum
+induced by the wave equation, which ranges over all the imaginary axis. This approach is strongly
+inspired by the methods introduced by D. Pelinovsky and M. Chugunova [9, 42, 43]. The workplan
+can be summarized as follows. It can be shown that the eigenproblem LX “ λX for L is equivalent
+to a generalized eigenvalue problem AW “ αKW, with α “ ´λ2, see Proposition 5.4 and 5.5
+below, where the auxiliary operators A and K depend on J , L . Then, we need to identify negative
+eigenvalues and complex but non real eigenvalues of the generalized eigenproblem. To this end, we
+appeal to a counting statement due to [9].
+5.2
+Spectral properties of the operator L
+The stability analysis relies on the spectral properties of L , collected in the following claim.
+Proposition 5.3 Let L the linear operator defined by (50) on DpL q Ă V . Suppose (9). Then,
+the following assertions hold:
+33
+
+1. σesspL q “
+␣ 1
+2, 2c2(
+,
+2. L has a finite number of negative eigenvalues, with eigenfunctions in DpL q, given by
+npL q
+“
+1 ` #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ă 0 and σ1,m “ 0u
+`#tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u.
+In particular, npL q “ 1 when k “ 0. The eigenspaces associated to the negative eigenvalues
+are all finite-dimensional.
+3. With X0 “ p0, 1, 0, 0q, we have spanRtX0u Ă KerpL q. Moreover, given k P Zd∖t0u, let K˚ “
+tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 “ 0 and σ1,m “ 0u. Then, we get dimpKerpL qq “ 1 ` #K˚.
+We remind the reader that σ1 is assumed radially symmetric, see (H1). Consequently σ1,m “
+σ1,´m “ σ1,˘m and both #K˚ and #tm P Zd ∖t0u, m4 ´4pk¨mq2 ď 0 and σ1,m � 0u are necessarily
+even.
+Proof.
+Since L is self-adjoint, σpL q Ă R. Let us study the eigenproblem for L : λX “ L X
+means
+$
+’
+’
+’
+’
+’
+’
+’
+’
+&
+’
+’
+’
+’
+’
+’
+’
+’
+%
+λq “ ´1
+2∆xq ` k ¨ ∇xp ` γσ1 ‹
+ˆˆ
+Rnp´∆q´1{2σ2φ dz
+˙
+,
+λp “ ´1
+2∆xp ´ k ¨ ∇xq,
+λφ “ 1
+2φ ` γp´∆q´1{2σ2σ1 ‹ q,
+λπ “ 2c2π.
+(52)
+Clearly λ “ 2c2 is an eigenvalue with eigenfunctions of the form p0, 0, 0, πq, π P L2pTd ˆ Rnq.
+As a consequence, dimpKerpL ´ 2c2Iqq is not finite and 2c2 P σesspL q.
+We turn to the case λ � 2c2, where the last equation imposes π “ 0. Using Fourier series, we
+obtain
+λqm “ m2
+2 qm ` ik ¨ mpm ` γp2πqdσ1,m
+ˆˆ
+Rnp´∆q´1{2σ2φm dz
+˙
+,
+λpm “ m2
+2 pm ´ ik ¨ mqm,
+λφm “ 1
+2φm ` γp2πqdp´∆q´1{2σ2σ1,mqm.
+(53)
+where qm, pm P C are the Fourier coefficients of q, p P L2pTdq while φmpzq “
+1
+p2πqd
+´
+Td φpx, zqe´im¨x dx
+for all z P Rn and φ P L2pTd; L2pRnqq.
+We split the discussion into several cases.
+Case m “ 0. For m “ 0, the equations (53) degenerate to
+λq0 “ γp2πqdσ1,0
+ˆˆ
+Rnp´∆q´1{2σ2φ0 dz
+˙
+,
+λp0 “ 0,
+´
+λ ´ 1
+2
+¯
+φ0 “ γp2πqdp´∆q´1{2σ2σ1,0q0.
+34
+
+Combining the first and the third equation yields
+λ
+´
+λ ´ 1
+2
+¯
+q0 “ γ2p2πq2dσ2
+1,0κq0,
+still with κ “
+´
+p´∆q´1σ2σ2 dz. It permits us to identify the following eigenvalues:
+• λ “ 0 is an eigenvalue associated to the eigenfunction p0, 1, 0, 0q,
+• since σ1,0 “
+1
+p2πqd
+´
+Td σ1 dx � 0, and p´∆q´1{2σ2 � 0, λ “ 1{2 is an eigenvalue associated to
+eigenfunctions p0, 0, φ, 0q, for any function z ÞÑ φpzq orthogonal to p´∆q´1{2σ2. As before,
+since dimpKerpL ´ 1
+2Iqq is not finite, 1
+2 P σesspL q.
+• the roots of
+λ
+´
+λ ´ 1
+2
+¯
+´ γ2p2πq2dσ2
+1,0κ “ λ2 ´ λ
+2 ´ γ2p2πq2dσ2
+1,0κ “ 0,
+provide two additional eigenvalues
+λ˘ “
+1{2 ˘
+b
+1{4 ` 4γ2p2πq2dσ2
+1,0κ
+2
+,
+associated to the eigenfunctions p1, 0, γp2πqdσ1,0p´∆q´1{2σ2
+λ˘´1{2
+, 0q, respectively.
+To sum up, the Fourier mode m “ 0 gives rise to two positive eigenvalues (1/2 and λ`), one
+negative eigenvalue (λ´) and the eigenvalue 0, the last two being both one-dimensional. It tells us
+that
+dimpKerpL qq ě 1 and npL q ě 1.
+Case m � 0 with σ1,m “ 0. In this case, the m-mode equations (53) for the particle and the
+wave are uncoupled
+pλ ´ 1{2qφm “ 0,
+pMm ´ λq
+ˆ
+qm
+pm
+˙
+“ 0
+where we have introduced the 2 ˆ 2 matrix
+Mm “
+ˆ
+m2{2
+ik ¨ m
+´ik ¨ m
+m2{2
+˙
+.
+(54)
+We identify the following eigenvalues:
+• λ “ 1{2 is an eigenvalue associated to the eigenfunction p0, 0, eim¨xφpzq, 0q, for any φ P L2pRnq.
+Once again, this tells us that 1
+2 P σesspL q.
+• the eigenvalues λ˘ “ m2˘2k¨m
+2
+P R of the 2 ˆ 2 matrix Mm, associated to the eigenfunctions
+peim¨x, ¯ieim¨x, 0, 0q, respectively. Since trpMmq ą 0, at most only one of these eigenvalues
+can be negative, which occurs when detpMmq “ m4
+4 ´ pk ¨ mq2 ă 0.
+Given k P Zd, we conclude this case by asserting
+npL q ě 1 ` #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ă 0, σ1,m “ 0u,
+35
+
+and
+dimpKerpL qq ě 1 ` #tm P Zd ∖ t0u, m2 “ ˘2k ¨ m, σ1,m “ 0u.
+Case m � 0 with σ1,m � 0. Again, we distinguish several subcases.
+• if λ “ 1{2, the third equation on (53) imposes qm “ 0, and we are led to
+1 ´ m2
+2
+pm “ 0,
+ik ¨ mpm ` γp2πqdσ1,m
+ˆˆ
+Rnp´∆q´1{2σ2φm dz
+˙
+“ 0.
+Thus, λ “ 1{2 is an eigenvalue associated to the eigenfunctions:
+p0, 0, eim¨xφpzq, 0q, for any function z ÞÑ φpzq orthogonal to p´∆q´1{2σ2,
+(we recover the same eigenfunctions as for the case m “ 0),
+p0, eim¨x, 0, 0q if k ¨ m “ 0, m2 “ 1,
+and
+´
+0, ´γp2πqdκσ1,m
+ik ¨ m
+eim¨x, p´∆q´1{2σ2pzqeim¨x, 0
+¯
+if k ¨ m � 0, m2 “ 1.
+• if λ “ m2
+2 � 1
+2, (53) becomes
+0 “ ik ¨ mpm ` γp2πqdσ1,m
+ˆˆ
+Rnp´∆q´1{2σ2φm dz
+˙
+,
+0 “ ´ik ¨ mqm,
+m2 ´ 1
+2
+φm “ γp2πqdp´∆q´1{2σ2σ1,mqm.
+There is no non-trivial solution when k ¨ m � 0. Otherwise, we see that λ “ m2{2 is an
+eigenvalue associated to the eigenfunctions: p0, eim¨x, 0, 0q
+• if λ � t1
+2, m2
+2 u, we set µ “ λ ´ m2
+2 . We see that a non trivial solution of (53) exists if its
+component qm does not vanish. We combine the equations in (53) to obtain
+Ppµqqm “ 0
+where P is the third order polynomial
+Ppµq “ µ3 ` bµ2 ` cµ ` d,
+b “ m2 ´ 1
+2
+ě 0,
+c “ ´ppk ¨ mq2 ` γ2κp2πq2dσ2
+1,mq ă 0,
+d “ ´pk ¨ mq2 m2 ´ 1
+2
+ď 0. .
+Observe that d “ ´pk ¨ mq2b and pk ¨ mq2 ă |c| ă pk ¨ mq2 ` 1
+4. We thus need to examine the
+roots of this polynomial. To this end, we compute the discriminant
+D “ 18bcd ´ 4b3d ` b2c2 ´ 4c3 ´ 27d2.
+A tedious, but elementary, computation allows us to reorganize terms as follows
+D
+“
+4pk ¨ mq2`
+pk ¨ mq2 ´ b2˘2 ` b2σ2
+1,mγp20pk ¨ mq2 ` γσ2
+1,mq
+`4pk ¨ mq2σ2
+1,mγp2pk ¨ mq2 ` γσ2
+1,mq ` 4σ2
+1,mγ
+`
+pk ¨ mq4 ` 2pk ¨ mq2σ2
+1,mγ ` σ4
+1,mγ2˘
+,
+36
+
+where we have set γ “ γ2κp2πq2d. Since σ1,m � 0, we thus have D ą 0 and P has 3 distinct
+real roots, µ1 ă µ2 ă µ3. In order to bring further information about the location of the roots,
+we observe that limµÑ˘8 Ppµq “ ˘8 while Pp0q “ d ď 0 and P 1p0q “ c ă 0. Moreover,
+studying the zeroes of P 1pµq “ 3µ2 ` 2bµ ` c, we see that µmax “ ´b´
+?
+b2´3c
+3
+ă 0 is a local
+maximum and µmin “ ´b`
+?
+b2´3c
+3
+ą 0 is a local minimum. Moreover, P 2pµq “ 6µ ` 2b,
+showing that P is convex on the domain p´pm2 ´ 1q{6, `8q, concave on p´8, ´pm2 ´ 1q{6q.
+A typical shape of the polynomial P is depicted in Figure 1. From this discussion, we infer
+µ1 ă µmax ă µ2 ď 0 ă µmin ă µ3.
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+-40
+-30
+-20
+-10
+0
+10
+20
+30
+40
+50
+P( )
+Figure 1: Typical graph for µ ÞÑ Ppµq, with its roots µ1 ă µ2 ă µ3 and local extrema µmax,
+µmin
+Coming back to the issue of counting the negative eigenvalues of L , we are thus wondering
+whether or not λj “ µj ` m2{2 is negative. We already know that µ3 ą µmin ą 0, hence
+µ3 ą ´m2{2 and we have at most 2 negative eigenvalues for each Fourier mode m � 0 such
+that σ1,m � 0. To decide how many negative eigenvalues should be counted, we look at the
+sign of Pp´m2{2q (see Fig. 1):
+i) if Pp´m2{2q ą 0 then µ1 ă ´m2{2 ă µ2,
+ii) if Pp´m2{2q “ 0 then either ´m2{2 ă µmax, in which case µ1 “ ´m2{2 ă µ2, or
+´m2{2 ą µmax, in which case µ2 “ ´m2{2 ą µ1,
+iii) if Pp´m2{2q ă 0 then either ´m2{2 ă µmax, in which case ´m2{2 ă µ1 ă µ2, or
+´m2{2 ą µmax, in which case µ1 ă µ2 ă ´m2{2.
+However, we remark that
+Pp´m2{2q
+“
+´m6
+8 ` m4pm2 ´ 1q
+8
+` m2
+2 ppk ¨ mq2 ` γσ2
+1,mq ´ m2 ´ 1
+2
+pk ¨ mq2
+“
+´m4
+8
+´
+1 ´ 4γσ2
+1,m
+m2
+¯
+` pk ¨ mq2
+2
+“ ´1
+8pm4 ´ 4pk ¨ mq2 ´ 4m2γσ2
+1,mq,
+(55)
+37
+
+where, by virtue of (9), m � 0 and σ1m � 0, 1 ą 4
+γσ2
+1,m
+m2
+ą 0.
+This can be combined together with
+P 1p´m2{2q “3m4
+4 ´ m2pm2 ´ 1q
+2
+´ pk ¨ mq2 ´ γσ2
+1,m “ m4
+4 ` m2
+2 ´ pk ¨ mq2 ´ γσ2
+1,m
+“1
+4
+`
+m4 ´ 4pk ¨ mq2 ´ 4m2γσ2
+1,m
+˘
+` m2γσ2
+1,m ` m2
+2 ´ γσ2
+1,m
+“ ´ 2Pp´m2{2q ` m2
+2 ` pm2 ´ 1qγσ2
+1,m ą ´2Pp´m2{2q.
+Finally,
+P 2p´m2{2q “ ´2m2 ´ 1 ă 0.
+As a consequence, Pp´m2{2q ă 0 implies P 1p´m2{2q ą 0, while P 2p´m2{2q ă 0. This shows
+that ´m2{2 ă µ1. Therefore, in case iii), the only remaining possibility is the situation where
+Pp´m2{2q ă 0 with ´m2{2 ă µ1 ă µ2. As a conclusion, if Pp´m2{2q ă 0, all eigenvalues λj
+are positive.
+Next, we claim that case ii) cannot occur. Indeed, Pp´m2{2q “ 0 if and only if
+pm2 ´ 2k ¨ mqpm2 ` 2k ¨ mq “ 4m2γσ2
+1,m.
+In particular, the term on the left hand side of this equality has to be positive.
+This is
+possible if and only if both factors, which belong to Z, are positive. In this case, according
+to the sign of k ¨ m, one of them is ě m2 so that
+m2 ď 4m2γσ2
+1,m.
+This contradicts the smallness condition (9). Note that Pp´m2{2q � 0 implies λj � 0, i.e.
+m-modes with m � 0 and σ1,m � 0 cannot generate elements of KerpL q.
+As a conclusion, negative eigenvalues only come from case i) and for each m-mode such that
+Pp´m2{2q ą 0 we have exactly one negative eigenvalue. Going back to (55), in this case, we
+have
+pm4 ´ 4pk ¨ mq2q “ pm2 ´ 2k ¨ mqpm2 ` 2k ¨ mq ă m24γσ2
+1,m ă m2
+owing to (9). This excludes the possibility that m4´4pk¨mq2 ą 0, since we noticed above that
+whenever this term is positive, it is ě m2. Hence, case i) holds if and only if m4´4pk¨mq2 ď 0.
+This ends the counting of the negative eigenvalues of L in Proposition 5.3. Note that the
+associated eigenspaces are spanned by
+´
+eim¨x, ´
+ik ¨ m
+λ ´ m2{2eim¨x, eim¨x σ1,mγp2πqdp´∆zq´1{2σ2
+λ ´ 1{2
+, 0
+¯
+.
+The discussion has permitted us to find the elements of KerpL q. To be specific, the equation
+38
+
+L X “ 0 yields π “ 0 and the following relations for the Fourier coefficients
+m2
+2 pm ´ ik ¨ mqm “ 0,
+φm
+2 ` p2πqdγp´∆q´1{2σ2σ1,mqm “ 0,
+m2
+2 qm ` ik ¨ mpm ` p2πqdγσ1,m
+ˆ
+p´∆q´1{2σ2φm dz “ 0.
+We have seen that the mode m “ 0 gives rise the eigenspace spanned by p0, 1, 0, 0q. For m � 0, ele-
+ments of KerpL q can be obtained only in the case σ1,m “ 0. Moreover, the condition m2 “ ˘2k ¨m
+has to be fulfilled. In such a case, peim¨x, ¯ieim¨x, 0, 0q P KerpL q.
+Finally, it remains to prove that σesspL q “
+␣1
+2, 2c2(
+. We have already noticed that
+␣ 1
+2, 2c2(
+Ă
+σesspL q. Suppose, by contradiction, that there exists λ P σesspL q ∖
+␣ 1
+2, 2c2(
+. Hence, by Weyl’s
+criterion [42, Theorem B.14], there exists a sequence pXνqνPN with Xν “ pqν, pν, φν, πνq P DpL q
+such that, as ν goes to 8,
+}pL ´ λIqXν} Ñ 0,
+}Xν} “ 1 and Xν á 0 weakly in V .
+(56)
+Since λ � 1
+2 and λ � 2c2, from (52) and (56) we have
+}πν}L2pTd;L2pRnqq Ñ 0 and φν “ ´
+ˆ1
+2 ´ λ
+˙´1
+γp´∆q´1{2σ2σ1 ‹ qν ` εν
+with εν P L2pTd; L2pRnqq such that limνÑ8 }εν}L2pTd;L2pRnqq “ 0. This leads to
+››››´1
+2∆xqν ´ λqν ` k ¨ ∇xpν ´
+γ2κ
+1{2 ´ λΣ ‹ qν ` γσ1 ‹
+ˆˆ
+Rnp´∆q´1{2σ2εν dz
+˙››››
+L2pTdq
+ÝÝÝÑ
+νÑ8 0,
+››››´1
+2∆xpν ´ λpν ´ k ¨ ∇xqν
+››››
+L2pTdq
+ÝÝÝÑ
+νÑ8 0.
+Using the fact that the sequence ppqν, pν, ενqqνPN is bounded in L2pTdq ˆ L2pTdq ˆ L2pTd; L2pRnqq,
+we deduce that pqν, pνqνPN is bounded in H2pTdq ˆ H2pTdq. Indeed, reasoning on Fourier series,
+this amounts to estimate
+ÿ
+mPZd
+|m|4p|qν,m|2 ` |pν,m|2q
+ď 2
+ÿ
+mPZd
+`
+|m2qν,m ` 2ik ¨ mpν,m|2 ` |m2pν,m ´ 2ik ¨ mqν,m|2q
+`8
+ÿ
+mPZd
+p|k ¨ mpν,m|2 ` |k ¨ mqν,m|2q
+ď 2
+›› ´ ∆xqν ` 2k ¨ ∇xpν
+››
+L2pTdq ` 2
+›› ´ ∆xpν ´ 2k ¨ ∇xqν
+››
+L2pTdq
+`4
+δ|k|4 ÿ
+mPZd
+`
+|qν,m|2 ` |pν,m|2˘
+` 4δ
+ÿ
+mPZd
+|m|4p|qν,m|2 ` |pν,m|2q.
+.
+Choosing 0 ă δ ă 1{4 and using the already known estimates, we conclude that }∆xqν}2
+L2 `
+}∆xpν}2
+L2 “ ř
+mPZd |m|4`
+|qν,m|2 ` |pν,m|2˘
+is bounded, uniformly with respect to ν. Hence, because
+of the compact Sobolev embedding of H2pTdq into L2pTdq, we have that pqν, pνqνPN has a (strongly)
+convergent subsequence in L2pTdq ˆ L2pTdq. As a consequence, the sequence pXνqνPN has a conver-
+39
+
+gent subsequence in V , which contradicts (56).
+A consequence of Proposition 5.3 is that 0 is an isolated eigenvalue of L . Since the restriction
+of L to the subspace pKerpL qqK is, by definition, injective, it makes sense to define on it its inverse
+L ´1, with domain RanpL q Ă pKerpL qqK Ă V . In fact, 0 being an isolated eigenvalue, RanpL q is
+closed and coincides with pKerpL qqK, [42, Section B.4]. This can be shown by means of spectral
+measures. Given X P pKerpL qqK, the support of the associated spectral measure dµX does not
+meet the interval p´ǫ, `ǫq for ǫ ą 0 small enough, independent of X. Accordingly, we get
+}L X}2 “
+ˆ `8
+´8
+λ2 dµXpλq “
+ˆ
+|λ|ěǫ
+λ2 dµXpλq ě ǫ2}X}2.
+In particular, the Fredholm alternative applies: for any Y P pKerpL qqK, there exists a unique
+X P pKerpL qqK such that L X “ Y . We will denote X “ L ´1Y .
+For further purposes, let us set
+X0 “ p0, 1, 0, 0q P KerpL q and Y0 “ ´J X0 “ p1, 0, 0, 0q.
+Note that Y0 P pKerpL qqK, so that it makes sense to consider the equation
+L U0 “ Y0.
+We find
+πm “ 0,
+φm “ ´2γp2πqdp´∆q´1{2σ2σ1,mqm,
+m2pm “ 2ik ¨ mqm,
+and
+m2qm ` 2ik ¨ mpm ` 2γp2πqdσ1,m
+ˆ
+p´∆q´1{2σ2φm dz “ δ0,m.
+It yields, for m � 0, pm4
+4 ´ pk ¨ mq2 ´ γ|σ1,m|2m2¯
+qm “ 0 and q0 “ ´
+1
+2γ2p2πq2d|σ1,0|2κ. Therefore, we
+can set
+U0 “ L ´1Y0 “ ´
+1
+2γ2p2πq2d|σ1,0|2κ
+`
+1, 0, ´2γp2πqdp´∆q´1{2σ2σ1,0, 0
+˘
+,
+solution of L U0 “ Y0 such that U0 P pKerpL qqK. We note that
+pU0, Y0q “ ´
+1
+2γ2p2πqd|σ1,0|2κ ă 0.
+(57)
+5.3
+Reformulation of the eigenvalue problem, and counting theo-
+rem
+The aim of the section is to introduce several reformulations of the eigenvalue problem. This will
+allow us to make use of general counting arguments, set up by [9, 42, 43].
+Proposition 5.4 Let us set M “ ´J L J . The coupled system
+M Y “ ´λX,
+L X “ λY,
+(58)
+admits a solution with λ � 0, X P DpL q ∖ t0u, Y P DpJ L J q ∖ t0u iff there exists two vectors
+X˘ P DpLq ∖ t0u that satisfy LX˘ “ ˘λX˘.
+40
+
+Let P stand for the orthogonal projection from V to pKerpL qqK Ă V .
+Proposition 5.5 Let us set A “ PM P and K “ PL ´1P. Let us define the following Hilbert
+space
+H “ DpM q X pKerpL qqK Ă V .
+The coupled system (58) has a pair of non trivial solutions p˘λ, X, ˘Y q, with λ � 0 iff the gener-
+alized eigenproblem
+AW “ αKW,
+W P H ,
+(59)
+admits the eigenvalue α “ ´λ2 � 0, with at least two linearly independent eigenfunctions.
+Recall that the plane wave solution obtained Section 2.1 is spectrally stable, if the spectrum
+of L is contained in iR. In view of Propositions 5.4 and 5.5, this happens if and only if all the
+eigenvalues of the generalized eigenproblem (59) are real and positive. In other words, the presence
+of spectrally unstable directions corresponds to the existence of negative eigenvalues or complex
+but non real eigenvalues of the generalized eigenproblem (59).
+Our goal is then to count the eigenvalues α of the generalized eigenvalue problem (59). In
+particular we define the following quantities:
+• N ´
+n , the number of negative eigenvalues
+• N 0
+n, the number of eigenvalues zero
+• N p
+n, the number of positive eigenvalues
+of (59), counted with their algebraic multiplicity, the eigenvectors of which are associated to non-
+positive values of the the quadratic form W ÞÑ pKW|Wq “ pL ´1PW|PWq. Moreover, let NC`
+be the number of eigenvalues α P C with Impαq ą 0.
+As pointed out above, the eigenvalues counted by N ´
+n and NC` correspond to cases of instabil-
+ities for the linearized problem (38). Note that to prove the spectral stability, it is enough to show
+that the generalized eigenproblem (59) does not have negative eigenvalues and NC` “ 0. Indeed,
+as a consequence of Propositions 5.4 and 5.5 and Lemma 5.1, if α P C ∖ R is an eigenvalue of (59),
+then ¯α is an eigenvalue too. Hence, if NC` “ 0, then the generalized eigenproblem (59) does not
+have solutions in C ∖ R.
+Finally, for using the counting argument introduce by Chugunova and Pelinovsky in [9], we need
+the following information on the essential spectrum of A, see [43, Lemma 2-(H1’) and Lemma 4].
+Lemma 5.6 Let M “ ´J L J be defined on V . Then σesspM q “ r0, `8q. Let A “ PM P and
+K “ PL ´1P be defined on H . Then σesspAq “ r0, `8q and we can find δ˚, d˚ ą 0 such that for
+any real number 0 ă δ ă δ˚, A`δK admits a bounded inverse and we have σesspA`δKq Ă rd˚δ, `8q.
+Proof.
+We check that
+J L J X “
+¨
+˚
+˚
+˚
+˚
+˚
+˚
+˝
+∆xq
+2
+´ k ¨ ∇xp
+∆xp
+2
+` k ¨ ∇xq ` γσ1 ‹
+ˆ
+p´∆zq´1{2σ2p´∆zq1{2π dz
+2c2∆zφ
+∆zπ
+2
+` γσ2σ1 ‹ p
+˛
+‹‹‹‹‹‹‚
+.
+41
+
+As a matter of fact, for any φ P H2pRnq, the vector Xe “ p0, 0, φ, 0q lies in pKerpL qqK and satisfies
+J L J Xe “
+¨
+˚
+˚
+˝
+0
+0
+2c2∆zφ
+0
+˛
+‹‹‚P pKerpL qqK.
+Consequently M Xe “ AXe “ ´J L J Xe “ p0, 0, ´2c2∆zφ, 0q. It indicates that a Weyl sequence
+for A ´ λI, λ ą 0, can be obtained by adapting a Weyl sequence for p´∆z ´ µIq, µ ą 0. Let us
+consider a sequence of smooth functions ζν P C8
+c pRnq such that supppζνq Ă Bp0, ν ` 1q, ζνpzq “ 1
+for x P Bp0, νq and }∇zζν}L8pRnq ď C0 ă 8, }D2
+zζν}L8pRnq ď C0 ă 8, uniformly with respect to
+ν P N. We set φνpzq “ ζνpzqeiξ¨z{p
+?
+2cq for some ξ P Rn. We get
+p´|ξ|2 ´ 2c2∆zqφνpzq “ eiξ¨z{p
+?
+2cq´ 2i
+?
+2cξ ¨ ∇zζν ` 2c2∆zζν
+¯
+pzq,
+which is thus bounded in L8pRnq and supported in Bp0, ν ` 1q ∖ Bp0, νq. It follows that }p´|ξ|2 ´
+2c2∆zqφν}2
+L2pRnq ≲ νn´1, while }φν}2
+L2pRnq ≳ νn. Accordingly, we obtain
+}φν}2
+L2pRnq
+}p´|ξ|2´2c2∆zqφν}2
+L2pRnq
+≳
+ν Ñ 8 as ν Ñ 8. Therefore, φν equally provides a Weyl sequence for M ´ |ξ|2I and A ´ |ξ|2I,
+showing the inclusions r0, 8q Ă σesspM q and r0, 8q Ă σesspAq.
+Next, let λ � r0, 8q. We suppose that we can find a Weyl sequence pXνqνPN for M , such that
+M Xν ´ λXν
+“
+¨
+˚
+˚
+˚
+˚
+˚
+˚
+˝
+´λqν ´ ∆xqν
+2
+` k ¨ ∇xpν
+´λpν ´ ∆xpν
+2
+´ k ¨ ∇xqν ´ γσ1 ‹
+ˆ
+p´∆zq´1{2σ2p´∆zq1{2πν dz
+´λφν ´ 2c2∆zφν
+´λπν ´ ∆zπν
+2
+´ γσ2σ1 ‹ pν
+˛
+‹‹‹‹‹‹‚
+“
+¨
+˚
+˚
+˝
+q1
+ν
+p1
+ν
+φ1
+ν
+π1
+ν
+˛
+‹‹‚ÝÝÝÑ
+νÑ8 0,
+with, moreover, }Xν} “ 1 and Xν á 0 weakly in V . In particular, we can set
+x
+φνpx, ξq “
+x
+φ1νpx, ξq
+2c2|ξ|2 ´ λ.
+(60)
+It defines a sequence which tends to 0 strongly L2pTd ˆ Rnq since, writing λ “ a ` ib P C ∖ r0, 8q,
+we get |2c2|ξ|2 ´λ|2 “ |2c2|ξ|2 ´a|2 `b2 which is ě b2 ą 0 when λ � R, and, in case b “ 0, ě a2 ą 0.
+Similarly, we can write
+x
+πνpx, ξq “
+2x
+π1νpx, ξq
+|ξ|2 ´ 2λ
+loooomoooon
+“hνpx,ξqPL2pTdˆRnq
+`γ
+2x
+σ2pξq
+|ξ|2 ´ 2λ
+loooomoooon
+PL2pRnq
+σ1 ‹ pν,
+(61)
+42
+
+where hν tends to 0 strongly L2pTd ˆ Rnq. We are led to the system
+¨
+˚
+˝
+´
+´
+λ ` ∆x
+2
+¯
+qν ` k ¨ ∇xpν
+´k ¨ ∇xqν ´
+´
+λ ` ∆x
+2
+¯
+pν ´ 2γ2
+ˆ
+|x
+σ2|2
+p2πqnp|ξ|2 ´ 2λq dξ ˆ Σ ‹ pν
+˛
+‹‚
+“
+¨
+˝
+q1
+ν
+p1
+ν ´ γσ1 ‹
+ˆ x
+σ2pξq
+|ξ| hνpx, ξq
+dξ
+p2πqn
+˛
+‚ÝÝÝÑ
+νÑ8 0.
+(62)
+Reasoning as in the proof of Proposition 5.3-1), we conclude that Xν converges strongly to 0
+in V , a contradiction.
+Hence, λ P C ∖ r0, 8q cannot belong to σesspM q and the identification
+σesspM q “ r0, 8q holds.
+Proposition 5.3-3) identifies KerpL q. Let us introduce the mapping
+Ă
+P :
+ˆq
+p
+˙
+P L2pTdqˆL2pTdq ÞÝÑ
+¨
+˚
+˚
+˝
+ÿ
+mPK˚, k¨mą0
+pqm ´ ipmqeim¨x `
+ÿ
+mPK˚, k¨mă0
+pqm ` ipmqeim¨x
+p0 ` i
+ÿ
+mPK˚, k¨mą0
+pqm ´ ipmqeim¨x ´ i
+ÿ
+mPK˚, k¨mă0
+pqm ` ipmqeim¨x
+˛
+‹‹‚.
+Then,
+X “
+¨
+˚
+˚
+˝
+q
+p
+φ
+π
+˛
+‹‹‚ÞÝÑ
+¨
+˚
+˚
+˝
+Ă
+P
+ˆ
+q
+p
+˙
+0
+0
+˛
+‹‹‚
+is the projection of V on KerpL q. Accordingly, we realize that P does not modify the last two
+components of a vector X “ pq, p, φ, πq P V , and for X P pKerpL qqK, we have p0 “ 0, and
+qm “ ˘ipm for any m P K˚, depending on the sign of k ¨ m.
+Now, let λ P C ∖ r0, 8q and suppose that we can exhibit a Weyl sequence pXνqνPN for A ´ λI:
+Xν P H Ă pKerpL qqK, PXν “ Xν, }Xν} “ 1, Xν á 0 in V and limνÑ8 }pA ´ λIqXν} “ 0. We
+can apply the same reasoning as before for the last two components of pA ´ λIqXν; it leads to (60)
+and (61), where, using λ � r0, 8q, φν and hν converge strongly to 0 in L2pTd ˆ Rnq. We arrive at
+the following analog to (62)
+pI ´ Ă
+Pq
+¨
+˚
+˝
+´
+´
+λ ` ∆x
+2
+¯
+qν ` k ¨ ∇xpν
+´k ¨ ∇xqν ´
+´
+λ ` ∆x
+2
+¯
+pν ´ 2γ2
+ˆ
+|x
+σ2|2
+p2πqnp|ξ|2 ´ 2λq dξ ˆ Σ ‹ pν
+˛
+‹‚
+“
+ˆq1
+ν
+p1
+ν
+˙
+´ pI ´ Ă
+Pq
+¨
+˝
+0
+γσ1 ‹
+ˆ x
+σ2pξq
+|ξ| hνpx, ξq
+dξ
+p2πqn
+˛
+‚ÝÝÝÑ
+νÑ8 0.
+(63)
+In order to derive from (63) an estimate in a positive Sobolev space as we did in the proof of
+Proposition 5.3-1), we should consider the Fourier coefficients arising from ´ 1
+2∆xqν ` k ¨ ∇xpν and
+´ 1
+2∆xpν ´k¨∇xqν, namely Qm “ m2
+2 qν,m`ik¨mpν,m and Pm “ m2
+2 pν,m´ik¨mqν,m. Only the coef-
+ficients belonging to K˚ are affected by the action of Ă
+P, which leads to Qm ´ pQm ¯ iPmq “ ˘iPm
+and Pm ¯ ipQm ¯ iPmq “ ¯iQm, according to the sign of k ¨ m. However, we bear in mind that
+qm “ ˘ipm when m P K˚ with ˘k ¨ m ą 0. Hence, for coefficients in K˚, the contributions of the
+differential operators reduces to ˘im2pm “ ˘m2qm and ¯im2qm “ ˘m2pm, respectively. Note
+43
+
+also that for these coefficients there is no contributions coming from the convolution with σ1 in
+(63) since σ1,m “ 0 for m P K˚. Therefore, reasoning as in the proof of Proposition 5.3-1) for
+coefficients m P Zd ∖ K˚, we can obtain a uniform bound on ř
+mPZd |m|4p|qν,m|2 ` |pν,m|2q, which
+provides a uniform H2 bound on qν and pν, leading eventually to a contradiction. We conclude
+that σesspAq “ r0, 8q.
+Let δ ą 0 and consider the shifted operator A ` δK. As a consequence of Lemma 5.10, we will
+see that KerpA ` δKq “ t0u for any δ ą 0: 0 is not an eigenvalue for A ` δK; let us justify it does
+not belong to the essential spectrum neither. To this end, we need to detail the expression of the
+operator K. Given X P H , we wish to find X1 P H satisfying
+L X1 “
+¨
+˚
+˚
+˚
+˚
+˚
+˚
+˝
+´1
+2∆xq1 ` k ¨ ∇xp1 ` γσ1 ‹
+ˆˆ
+Rnp´∆q´1{2σ2φ1 dz
+˙
+´1
+2∆xp1 ´ k ¨ ∇xq1
+1
+2φ1 ` γp´∆q´1{2σ2σ1 ‹ q1
+2c2π1
+˛
+‹‹‹‹‹‹‚
+“ X.
+We infer π1 “
+π
+2c2 and the relation φ1 “ 2φ ´ 2γp´∆zq´1{2σ2σ1 ‹ q1. In turn, the Fourier coefficients
+of q1, p1 are required to satisfy
+ˆ
+m2{2 ´ 2γ2κp2πq2d|σ1,m|2
+ik ¨ m
+´ik ¨ m
+m2{2
+˙ ˆ
+q1
+m
+p1
+m
+˙
+“
+¨
+˝qm ´ 2γp2πqdσ1,m
+ˆ
+p´∆q´1{2σ2φm dz
+pm
+˛
+‚.
+When m � 0, m � K˚, the matrix of this system has its determinant equal to
+det “ m4
+4
+`
+1 ´ 4γ2κp2πq2d |σ1,m|2
+m2
+˘
+´ pk ¨ mq2.
+Owing to (9), since pk ¨ mq2 takes values in N, it does not vanish and we obtain q1
+m, p1
+m by solving
+the system
+q1
+m “
+1
+det
+ˆm2
+2
+´
+qm ´ 2γp2πqdσ1,m
+ˆ
+p´∆q´1{2σ2φm dz
+¯
+´ ik ¨ mpm
+˙
+,
+p1
+m “
+1
+det
+ˆ
+`ik ¨ m
+´
+qm ´ 2γp2πqdσ1,m
+ˆ
+p´∆q´1{2σ2φm dz
+¯
+`
+´m2
+2 ´ 2γ2κp2πq2d|σ1,m|2¯
+pm
+˙
+.
+If m P K˚ we find a solution in pKerpL qqK by setting p1
+m “ pm
+m2 , q1
+m “ ˘ip1
+m, according to the sign
+of k ¨m; if m “ 0, we set p1
+0 “ 0 and q1
+0 “
+1
+2γ2κp2πq2d|σ1,0|2
+`
+q0 ´2γp2πqdσ1,0
+´
+p´∆q´1{2σ2φ0 dz
+˘
+. This
+defines X1 “ KX.
+Therefore, the last two components of pA ` δK ´ λIqX read
+p2δ ´ λqφ ´ 2c2∆zφ ´ 2δγp´∆q´1{2σ2σ1 ‹ q1,
+´ δ
+2c2 ´ λ
+¯
+π ´ 1
+2∆zπ ´ γσ2σ1 ‹ p1.
+Hence, when λ does not belong to rδd˚, 8q, with d˚ “ minp2,
+1
+2c2q, we can repeat the analysis
+performed above to establish that λ � σesspA ` δKq. In particular the essential spectrum of A has
+been shifted away from 0.
+We are now able to apply the results of Chugunova and Pelinovsky [9] (see also [43]), to obtain
+44
+
+the following.
+Theorem 5.7 [9, Theorem 1] Let L be defined by (50). Suppose (9). With the operators M , A, K
+defined as in Propositions 5.4-5.5, the following identity holds
+N ´
+n ` N 0
+n ` N `
+n ` NC` “ npL q.
+Let us now detail the proof of Proposition 5.4 and 5.5, adapted from [43, Prop. 1 & Prop. 3].
+Proof of Propositions 5.4 and 5.5.
+The goal is to establish connections between the following
+three problems:
+(Ev) the eigenvalue problem LX “ λX, with L “ J L ,
+(Co) the coupled problem L X “ λY , M Y “ ´λX, with M “ ´J L J ,
+(GEv) the generalized eigenvalue problem AW “ αKW, with A “ PM P, K “ PL ´1P, the
+projection P on pKerpL qqK, and W P H “ DpM q X pKerpL qqK.
+The proof of Propositions 5.4 and 5.5 follows from the following sequence of arguments.
+(i) By Lemma 5.1, we already know that if there exists a solution pλ, X`q of (Ev), with λ � 0
+and X` � 0, then, there exists X´ � 0, such that p´λ, X´q satisfies (Ev). Being eigenvectors
+associated to distinct eigenvalues, X` and X´ are linearly independent. Note that only this
+part of the proof uses the specific structure of the operator L.
+(ii) From these eigenpairs for L, we set
+X “ X` ` X´
+2
+,
+Y “
+Ă
+J
+ˆX` ´ X´
+2
+˙
+.
+Since X` and X´ are linearly independent, we have X � 0, Y � 0. Moreover, X “ X``X´
+2
+and J Y “ X`´X´
+2
+are linearly independent. We get
+L X “
+Ă
+J LX “
+Ă
+J
+ˆλ
+2pX` ´ X´q
+˙
+“ λY,
+M Y “ ´J L
+ˆX` ´ X´
+2
+˙
+“ ´L
+ˆX` ´ X´
+2
+˙
+“ ´λ
+2pX` ` X´q “ ´λX,
+so that pλ, X, Y q satisfies (Co).
+(iii) If pλ, X, Y q is a solution (Co), then p´λ, X, ´Y q satisfies (Co) too.
+(iv) Let pλ, X, Y q be a solution (Co). Set
+X1 “ J Y,
+Y 1 “
+Ă
+J X.
+We observe that
+M Y 1 “ ´J L J Ă
+J X “ ´J L X “ ´J pλY q “ ´λX1,
+L X1 “ L J Y “
+Ă
+J J L J Y “ ´ Ă
+J M Y “ λ Ă
+J X “ λY 1,
+which means that pλ, J Y, Ă
+J Xq is a solution of (Co). Moreover, if X and J Y are linearly
+independent, Y and
+Ă
+J X are linearly independent too.
+45
+
+(v) Let pλ, X, Y q be a solution (Co), with X � 0. We get
+LpX ˘ J Y q
+“
+J L X ˘ J L J Y “ J L X ¯ M Y
+“
+J pλY q ˘ λX “ ˘λpX ˘ J Y q,
+so that p˘λ, X ˘ J Y q satisfy (Ev). In the situation where X and J Y are linearly inde-
+pendent, we have X ˘ J Y � 0 and p˘λ, X ˘ J Y q are eigenpairs for L. Otherwise, one of
+the vectors X ˘ J Y might vanish. Nevertheless, since only one of these two vectors can be
+0, we still obtain an eigenvector X˘ � 0 of L, associated to either ˘λ. Coming back to i), we
+conclude that ¯λ is an eigenvalue too.
+Items i) to v) justify the equivalence stated in Proposition 5.4.
+(vi) Let pλ, X, Y q be a solution (Co). From L X “ λY , we infer Y P RanpL q Ă pKerpL qqK so
+that PY “ Y . The relation thus recasts as
+X “ λPL ´1PY ` ˜Y,
+˜Y P KerpL q,
+P ˜Y “ 0.
+(Here, PL ´1PY stands for the unique solution of L Z “ Y which lies in pKerpL qqK.) We
+obtain
+PM Y
+“
+Pp ´ λXq “ ´λPpλPL ´1PY ` ˜Y q
+“
+´λ2PL ´1PY “ ´λ2KY “ PM PY “ AY,
+so that p´λ2, Y q satisfies (GEv). Going back to iv), we know that p´λ2, Ă
+J Xq is equally a
+solution to (GEv). If X and J Y are linearly independent, we obtain this way two linearly
+independent vectors, Y and
+Ă
+J X, solutions of (GEv) with α “ ´λ2.
+(vii) Let pα, Wq satisfy (GEv), with α � 0, W � 0. We set X “ ´M W
+?´α . We have
+Ă
+J X “ ´
+1
+?´α
+Ă
+J M W “
+1
+?´α
+Ă
+J J L J W “
+1
+?´αL J W
+which lies in RanpL q Ă pKerpL qqK. Thus, using P Ă
+J X “
+Ă
+J X, we compute
+K Ă
+J X “ PL ´1P Ă
+J X “ PL ´1 Ă
+J X “
+1
+?´αPL ´1L J W “
+1
+?´αPJ W.
+Next, we observe that
+A Ă
+J X “ PM P Ă
+J X “ ´PJ L J Ă
+J X “ ´PJ L X “
+1
+?´αPJ L M W.
+However, we can use PW “ W (since W P H Ă pKerpL qqK) and the fact that, for any
+vector Z, L Z “ L pI ´ PqZ ` L PZ “ 0 ` L PZ, which yields
+A Ă
+J X
+“
+1
+?´αPJ L PM PW “
+1
+?´αPJ L AW “ ´
+?
+´αPJ L KW
+“
+´ ?´αPJ L PL ´1PW “ ´ ?´αPJ L L ´1W “ ´ ?´αPJ W.
+We conclude that A Ă
+J X “ αK Ă
+J X: pα, Ă
+J Xq satisfies (GEv).
+(viii) Let pα, Wq satisfy (GEv), with α � 0, W � 0. We have
+PpM PW ´ αL ´1PWq “ 0
+and thus
+M PW ´ αL ´1PW “ ˜Y P KerpL q.
+46
+
+Let us set
+Y “ PW P pKerpL qqK,
+X “ ´M PW
+?´α
+“
+´1
+?´αp˜Y ` αL ´1PWq,
+so that
+L X “
+?
+´αPW “
+?
+´αY,
+M Y “ M PW “ ´
+?
+´αX.
+Therefore p ?´α, X, Y q satisfies (Co). By v), p˘ ?´α, X ˘ J Y q satisfy (Ev), and at least
+one of the vectors X ˘J Y does not vanish; using i), we thus obtain eigenpairs p˘ ?´α, X˘q
+of L. With ii), we construct solutions of (Co) under the form
+` ?´α, X``X´
+2
+, Ă
+J
+` X`´X´
+2
+˘˘
+,
+which, owing to iv) and vi), provide the linearly independent solutions
+`
+α, Ă
+J
+`X`˘X´
+2
+˘˘
+of
+(GEv). The dimension of the linear space of solutions of (GEv) is at least 2.
+At least one of these vectors X˘ is given by the formula
+˜X˘ “ ´ M W
+?´α ˘ J W.
+By the way, we indeed note that AW “ αKW, with W P H , can be cast as L J L J W “
+´αW (see Lemma 5.8 below) so that
+L
+´
+´ M W
+?´α ˘ J W
+¯
+“
+1
+?´αJ pL J L J Wq ˘ J L J W
+“
+?´αJ W ¯ M W “ ˘ ?´α
+´
+´ M W
+?´α ˘ J W
+¯
+.
+With these manipulations we have checked that p˘ ?´α, ˜X˘q satisfy (Ev). If both vectors
+˜X˘ are non zero, we get X˘ “ ˜X˘ and we recover W “
+Ă
+J
+` X`´X´
+2
+˘
+. If ˜X˘ “ 0, then, we
+get ˜X¯ “ ¯J W � 0, and we directly obtain X¯ “ ˜X¯, W “ ¯ Ă
+J X¯. In any cases, W lies
+in the space spanned by X` and X´ and the dimension of the space of solutions of (GEv)
+is even.
+This ends the proof of Proposition 5.4 and 5.5.
+5.4
+Spectral instability
+We are going to compute the terms arising in Theorem 5.7. Eventually, it will allow us to identify the
+possible unstable modes. In what follows, we find convenient to work with the operator M ´αL ´1
+instead of PpM ´ αL ´1qP “ A ´ αK, owing to to the following claim.
+Lemma 5.8 Let α � 0. In the space H “ DpM q X pKerpL qqK, the two subspaces KerpA ´ αKq
+and KerpM ´ αL ´1q coincide.
+Proof.
+Let X P H satisfy M X “ αL ´1X. Then, we have X “ PX and, thus, pA ´ αKqX “
+PpM ´ αL ´1qPX “ PpM X ´ αL ´1Xq “ 0, showing the inclusion KerpM ´ αL ´1q X H Ă
+KerpA ´ αKq.
+Conversely, the equation pA ´ αKqX “ 0, with X “ PX P pKerpL qqK means that pM ´
+αL ´1qX “ Y P KerpL q. Applying L then yields L M X “ αX. Since both terms of this relation
+lie in pKerpL qqK, it is legitimate to apply L ´1, showing that M X “ αL ´1X: we have shown
+KerpA ´ αKq X H Ă KerpM ´ αL ´1q.
+47
+
+Therefore, we shall consider the solutions of the generalized eigenvalue problem M X “ αL ´1X,
+with X P H . We rewrite the equation by introducing an auxiliary unknown:
+M X “ α ˜X,
+L ˜X “ X.
+Lemma 5.9 Suppose (9). N 0
+n “ 1.
+Proof.
+We are interested in the solutions of
+´1
+2∆xq ` k ¨ ∇xp “ 0,
+´1
+2∆xp ´ k ¨ ∇xq ´ γσ1 ‹
+ˆ
+σ2π dz “ 0,
+´2c2∆zφ “ 0,
+´1
+2∆zπ ´ γσ2σ1 ‹ p “ 0.
+We infer φpx, zq “ 0 and pπpx, ξq “ 2γ x
+σ2pξq
+|ξ|2 σ1 ‹ ppxq, and, next,
+´1
+2∆xq ` k ¨ ∇xp “ 0,
+´1
+2∆xp ´ k ¨ ∇xq ´ 2γ2κΣ ‹ p “ 0
+with Σ “ σ1 ‹ σ1. In terms of Fourier coefficients, it becomes
+m2
+2 qm ` ik ¨ mpm “ 0,
+m2
+2 pm ´ ik ¨ mqm ´ 2p2πq2dγ2κ|σ1,m|2pm “ 0.
+For m “ 0, we get p0 “ 0 and we find the eigenfunction p1, 0, 0, 0q “ Y0 “ ´J X0 with X0 “
+p0, 1, 0, 0q P KerpL q.
+For m � 0 with σ1,m � 0, we get
+m4 ´ 4pk ¨ mq2 “ 2p2πq2dγ2κ|σ1,m|2
+loooooooooomoooooooooon
+Pp0,1q
+m2.
+which cannot hold (see the proof of Proposition 5.3 for more details).
+For m � 0 with σ1,m “ 0, we get Mm
+ˆqm
+pm
+˙
+“ 0 with Mm defined in (54).
+As far as
+m4 ´4pk ¨mq2 � 0, Mm is invertible and the only solution is pm “ 0 “ qm. If m4 ´4pk ¨mq2 “ 0, we
+find the eigenfunctions peik¨m, ˘ieik¨m, 0, 0q. These functions belong to KerpL q, and thus do not
+lie in the working space H .
+We conclude that KerpM q “ spanRtY0u. Moreover, this vector Y0 does not belong to RanpM q
+so that the algebraic multiplicity of the eigenvalue 0 is 1. Finally, bearing in mind (57), which can
+be recast as pKY0|Y0q ă 0, we arrive at N 0
+n “ 1.
+Lemma 5.10 Suppose (9). The generalized eigenproblem (59) does not admit negative eigenvalues.
+In particular, N ´
+n “ 0.
+48
+
+Proof.
+Let α ă 0, X “ pq, p, φ, πq and ˜X “ p˜q, ˜p, ˜φ, ˜πq satisfy
+´1
+2∆xq ` k ¨ ∇xp “ α˜q,
+´1
+2∆xp ´ k ¨ ∇xq ´ γσ1 ‹
+ˆ
+σ2π dz “ α˜p,
+´2c2∆zφ “ α˜φ,
+´1
+2∆zπ ´ γσ2σ1 ‹ p “ α˜π,
+(64)
+where
+q “ ´1
+2∆x˜q ` k ¨ ∇x˜p ` γσ1 ‹
+ˆ
+p´∆zq´1{2σ2 ˜φ dz,
+p “ ´1
+2∆x˜p ´ k ¨ ∇x˜q,
+φ “ 1
+2
+˜φ ` γp´∆zq´1{2σ2σ1 ‹ ˜q,
+π “ 2c2˜π.
+(65)
+This leads to solve an elliptic equation for π
+´|α|
+c2 ´ ∆z
+¯
+π “ 2γσ2σ1 ‹ p.
+In other words, we get, by means of Fourier transform
+pπpx, ξq “ 2γσ1 ‹ ppxq ˆ
+x
+σ2pξq
+|ξ|2 ` |α|{c2 .
+On the same token, we obtain
+´|α|
+c2 ´ ∆z
+¯
+˜φ “ ´2γp´∆zq1{2σ2σ1 ‹ ˜q,
+which yields
+p˜φpx, ξq “ ´2γσ1 ‹ ˜qpxq ˆ
+|ξ|x
+σ2pξq
+|ξ|2 ` |α|{c2 .
+For λ ą 0, we introduce the symbol
+0 ď κλ “
+ˆ |x
+σ2pξq|2
+|ξ|2 ` λ ď κ.
+It turns out that
+´1
+2∆xq ` k ¨ ∇xp “ α˜q,
+´1
+2∆xp ´ k ¨ ∇xq ´ 2γ2κ|α|{c2Σ ‹ p “ α˜p,
+with
+q “ ´1
+2∆x˜q ` k ¨ ∇x˜p ´ 2γ2κ|α|{c2Σ ‹ ˜q,
+p “ ´1
+2∆x˜p ´ k ¨ ∇x˜q.
+49
+
+For the Fourier coefficients, it casts as
+m2
+2 qm ` ik ¨ mpm “ α˜qm,
+m2
+2 pm ´ ik ¨ mqm ´ 2γ2κ|α|{c2p2πq2d|σ1,m|2pm “ α˜pm,
+with
+qm “ m2
+2 ˜qm ` ik ¨ m˜pm ´ 2γ2κ|α|{c2p2πq2d|σ1,m|2˜qm,
+pm “ m2
+2 ˜pm ´ ik ¨ m˜qm.
+We are going to see that these equations do not have non trivial solutions with α ă 0:
+• If m “ 0, we get p0 “ 0, ˜q0 “ 0, and, consequently, ˜p0 “ 0, q0 “ 0. Hence, for α ă 0, we
+cannot find an eigenvector with a non trivial 0-mode.
+• If m � 0 and σ1,m “ 0, we see that pqm, pmq and p˜qm, ˜pmq are related by
+Mm
+ˆqm
+pm
+˙
+“ α
+ˆ˜qm
+˜pm
+˙
+,
+ˆqm
+pm
+˙
+“ Mm
+ˆ˜qm
+˜pm
+˙
+.
+(66)
+It means that α is an eigenvalue of
+M2
+m “
+˜
+m4
+4 ` pk ¨ mq2
+im2k ¨ m
+´im2k ¨ m
+m4
+4 ` pk ¨ mq2
+¸
+.
+The roots of the characteristic polynomial of M2
+m are pm2
+2 ˘ k ¨ mq2 ě 0, which contradicts
+the assumption α ă 0.
+• For the case where m � 0 and σ1,m � 0, we introduce the shorthand notation am “
+2γ2p2πq2d|σ1,m|2κ|α|{c2, bearing in mind that 0 ă am ă m2
+2 by virtue of the smallness condi-
+tion (9). We are led to the systems
+ˆ
+Mm ´
+ˆ
+0
+0
+0
+am
+˙˙ ˆ
+qm
+pm
+˙
+“ α
+ˆ
+˜qm
+˜pm
+˙
+,
+ˆ
+qm
+pm
+˙
+“
+ˆ
+Mm ´
+ˆ
+am
+0
+0
+0
+˙˙ ˆ
+˜qm
+˜pm
+˙
+,
+which imply that α is an eigenvalue of the matrix
+ˆ
+Mm ´
+ˆ
+0
+0
+0
+am
+˙˙ ˆ
+Mm ´
+ˆ
+am
+0
+0
+0
+˙˙
+.
+However the eigenvalues of this matrix read
+` b
+m2
+2 pm2
+2 ´ amq ˘ pk ¨ mq2˘2 ě 0, contradicting
+that α is negative.
+Lemma 5.11 Suppose (9). N `
+n “ #tm P Zd ∖ t0u, σ1,m “ 0, and m4 ´ 4pk ¨ mq2 ă 0u.
+50
+
+Proof.
+We should consider the system of equations (64)-(65), now with α ą 0. For Fourier
+coefficients, it casts as
+m2
+2 qm ` ik ¨ mpm “ α˜qm,
+m2
+2 pm ´ ik ¨ mqm ´ γp2πqdσ1,m
+ˆ
+σ2πm dz “ α˜pm,
+´2c2∆zφm “ α˜φm,
+´1
+2∆zπm ´ γp2πqdσ1,mσ2pm “ α˜πm,
+where
+qm “ m2
+2 ˜qm ` ik ¨ m˜pm ` γp2πqdσ1,m
+ˆ
+p´∆zq´1{2σ2 ˜φm dz,
+pm “ m2
+2 ˜pm ´ ik ¨ m˜qm,
+φm “ 1
+2
+˜φm ` γp2πqdp´∆zq´1{2σ2σ1,m˜qm,
+πm “ 2c2˜πm.
+• For m “ 0, we obtain p0 “ 0, ˜q0 “ 0. Hence π0 satisfies p´α{c2 ´ ∆zqπ0 “ 0. Here, `α{c2
+lies in the essential spectrum of ´∆z and the only solution in L2 of this equation is π0 “ 0.
+In turn, this implies ˜p0 “ 0, p´α{c2 ´ ∆zqφ0 “ 0, and thus φ0 “ 0, q0 “ 0. Hence, for α ą 0,
+we cannot find an eigenvector with a non trivial 0-mode.
+• When m � 0 and σ1,m “ 0, we are led to p´α{c2 ´ ∆qφm “ 0, p´α{c2 ´ ∆qπm “ 0 that
+imply φm “ 0, πm “ 0.
+In turn, we get (66) for qm, pm, ˜qm, ˜pm.
+This holds iff α is an
+eigenvalue of M2
+m. If m4 � 4pk ¨ mq2, we find two eigenvalues αm,˘ “ pm2
+2 ˘ k ¨ mq2 ą 0, with
+associated eigenvectors Xm,˘ “ peim¨x, ¯ieim¨x, 0, 0q, respectively. To decide whether these
+modes should be counted, we need to evaluate the sign of pL ´1Xm,˘|Xm,˘q. We start by
+solving L X1
+m,˘ “ Xm,˘. It yields
+φ1
+m,˘
+2
+“ 0, 2c2π1
+m,˘ “ 0 and
+Mm
+ˆq1
+m,˘
+p1
+m,˘
+˙
+“
+ˆ 1
+¯i
+˙
+.
+We obtain
+q1
+m,˘ “
+2
+m2 ˘ 2k ¨ m,
+π1
+m,˘ “
+¯2i
+m2 ˘ 2k ¨ m,
+so that
+pL ´1Xm,˘|Xm,˘q
+“
+2
+m2 ˘ 2k ¨ m
+ˆˆ
+Td eim¨xe´im¨x dx `
+ˆ
+Tdp¯iqeim¨x˘ie´im¨x dx
+˙
+“
+4p2πqd
+m2 ˘ 2k ¨ m,
+the sign of which is determined by the sign of m2 ˘2k¨m. We count only the situation where
+these quantities are negative; reproducing a discussion made in the proof of Proposition 5.3,
+we conclude that
+N `
+n ě #tm P Zd ∖ t0u, σ1,m “ 0 and m4 ´ 4pk ¨ mq2 ă 0u.
+51
+
+When m4 ´ 4pk ¨ mq2 “ 0, the eigenvalues of M2
+n are 0 and m4, and we just have to consider
+the positive eigenvalue α “ m4, associated to the eigenvector Xm “ peim¨x, ˘ieim¨x, 0, 0q
+(depending whether m2
+2
+“ ¯k ¨ m). The equation L Ym “ Xm
+has infinitely many solu-
+tions of the form p2{m2eim¨x, 0, 0, 0q ` zp˘ieim¨x, eim¨x, 0, 0q, with z P C. We deduce that
+pL ´1Xm|Xmq “ 2p2πqd
+m2
+ą 0. Thus these modes do not affect the counting.
+• When m � 0 and σ1,m � 0, we are led to the relations p´α{c2 ´ ∆zqπm “ 2σ2γp2πqdσ1,mpm,
+p´α{c2 ´∆zq˜φm “ ´2p´∆zq1{2σ2γp2πqdσ1,m˜qm. The only solutions with square integrability
+on Rn are πm “ 0, ˜φm “ 0, pm “ 0, ˜qm “ 0. This can be seen by means of Fourier transform:
+p´α{c2 ´ ∆zqφ “ σ amounts to pφpξq “
+pσpξq
+|ξ|2´α{c2; due to (H4) this function has a singularity
+which cannot be square-integrable. In turn, this equally implies φm “ 0 and ˜πm “ 0. Hence,
+we arrive at m2
+2 qm “ 0 and ´ik ¨ mqm “ α˜pm, together with qm “ ik ¨ m˜pm and m2
+2 ˜pm “ 0.
+We conclude that α ą 0 cannot be an eigenvalue associated to a m-mode such that m � 0
+and σ1,m � 0.
+We can now make use of Theorem 5.7, together with Proposition 5.3. This leads to
+0 ` 1 ` #tm P Zd ∖ t0u, σ1,m “ 0, and m4 ´ 4pk ¨ mq2 ă 0u ` NC` “ N ´
+n ` N 0
+n ` N `
+n ` NC`
+“ npL q “ 1 ` #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ă 0 and σ1,m “ 0u
+`#tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u
+so that
+NC` “ #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u.
+Since the eigenvalue problem (59) does not have negative (real) eigenvalues, this is the only source
+of instabilities.
+As a matter of fact, when k “ 0, we obtain NC` “ 0, which yields the following statement,
+(hopefully!) consistent with Lemma 4.1 and Proposition 4.2.
+Corollary 5.12 Let k “ 0 and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose
+(9) holds. Then the plane wave solution peiωt1pxq, ´γΓpzq
+@
+σ
+D
+Td, 0q is spectrally stable.
+In contrast to what happens for the Hartree equation, for which the eigenvalues are purely
+imaginary, see Lemma 3.2, we can find unstable modes, despite the smallness condition (9). Let us
+consider the following two examples in dimension d “ 1, with k P Z ∖ t0u.
+Example 5.13 Suppose σ1,0 � 0, and σ1,1 � 0.
+Then, the set tm P Z ∖ t0u, m4 ´ 4k2m2 ď
+0 and σ1,m � 0u contains t´1, `1u (since 4k2 ě 1). Let k P Z ∖ t0u and ω ą 0 such that the
+dispersion relation (12) is satisfied. Then the plane wave solution peiωteikx, ´γΓpzq
+@
+σ1
+D
+Td, 0q is
+spectrally unstable.
+Example 5.14 Let m˚ P Z ∖ t0u be the first Fourier mode such that σ1,m˚ � 0. Let k P Z and
+ω ą 0 such that the dispersion relation (12) is satisfied. Then, for all k P Z such that 4k2 ă m2
+˚,
+the plane wave solution peiωteikx, ´γΓpzq
+@
+σ
+D
+Td, 0q is spectrally stable, while for all k P Z such that
+4k2 ě m2
+˚, the plane wave solution peiωteikx, ´γΓpzq
+@
+σ1
+D
+Td, 0q is spectrally unstable.
+52
+
+In general, if k P Zd ∖ t0u, the set tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u contains ´k
+and k provided σ1,k � 0. Hence, we have the following result.
+Corollary 5.15 Let k P Zd ∖ t0u and ω ą 0 such that the dispersion relation (12) is satis-
+fied.
+Suppose (9) holds and σ1,m � 0 for all m P Zd ∖ t0u.
+Then the plane wave solution
+peipωt`k¨xq, ´γΓpzq
+@
+σ1
+D
+Td, 0q is spectrally unstable.
+Remark 5.16 (Orbital instability) Given Corollary 5.15, it is natural to ask whether in this
+case the plane wave solution peipωt`k¨xq, ´γΓpzq
+@
+σ1
+D
+Td, 0q is orbitally unstable.
+Note that, if σ1,m � 0 for all m P Zd ∖ t0u, we deduce from Proposition 5.3 that npLq ě
+3.
+As a consequence, the arguments used in [22] to prove the orbital instability (see also [38,
+41]) do not apply. It seems then necessary to work directly with the propagator generated by the
+linearized operator as in [23, 16]. In particular, one has to establish Strichartz type estimates for
+the propagator of L (a task we leave for future work).
+A
+Scaling of the model and physical interpretation
+It is worthwhile to discuss the meaning of the parameters that govern the equations and the
+asymptotic issues. Going back to physical units, the system reads
+ˆ
+iℏBtU ` ℏ2
+2m∆xU
+˙
+pt, xq “
+ˆˆ
+TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dy dz
+˙
+upt, xq,
+(67a)
+pB2
+ttΨ ´ κ2∆zΨqpt, x, zq “ ´σ2pzq
+ˆˆ
+Td σ1px ´ yq|Upt, yq|2 dy
+˙
+.
+(67b)
+The quantum particle is described by the wave function pt, xq ÞÑ Upt, xq: given Ω Ă Td, the integral
+´
+Ω |Upt, xq|2 dx gives the probability of presence of the quantum particle at time t in the domain
+Ω; this is a dimensionless quantity. In (67a), ℏ stands for the Planck constant; its homogeneity
+is MassˆLength2
+Time
+(and its value is 1.055 ˆ 10´34 Js) and m is the inertial mass of the particle. Let
+us introduce mass, length and time units of observations: M, L and T. It helps the intuition to
+think of the z directions as homogeneous to a length, but in fact this is not necessarily the case:
+we denote by Ψ and Z the (unspecified) units for Ψ and the zj’s. Hence, κ is homogeneous to the
+ratio Z
+T. The coupling between the vibrational field and the particle is driven by the product of
+the form functions σ1σ2, which has the same homogeneity as
+ℏ
+TΨLdZn from (67a) and as
+Ψ
+LdT2 from
+(67b), both are thus measured with the same units. From now on, we denote by ς this coupling
+unit. Therefore, we are led to the following dimensionless quantities
+U1pt1, x1q “ Upt1T, x1Lq
+c
+Ld m
+M,
+Ψ1pt1, x1, z1q “ 1
+ΨΨpt1T, x1L, z1Zq,
+σ1
+1px1qσ2pz1q “ 1
+ς σ1px1Lqσ2pz1Zq.
+Bearing in mind that u is a probability density, we note that
+ˆ
+Td |U1pt1, x1q|2 dx1 “ m
+M.
+53
+
+Dropping the primes, (67a)-(67b) becomes, in dimensionless form,
+ˆ
+iBtU ` ℏT
+mL2
+1
+2∆xU
+˙
+pt, xq “ ςΨLdZnT
+ℏ
+ˆˆ
+TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dy dz
+˙
+Upt, xq,
+(68a)
+´
+B2
+ttΨ ´ κ2T2
+Z2 ∆zΨ
+¯
+pt, x, zq “ ´ςT2
+Ψ
+M
+mσ2pzq
+ˆˆ
+Td σ1px ´ yq|Upt, yq|2 dy
+˙
+.
+(68b)
+Energy conservation plays a central role in the analysis of the system: the total energy is defined
+by using the reference units and we obtain
+E0 “
+´ ℏT
+mL2
+¯2 1
+2
+ˆ
+Td |∇xU|2 dx ` Ψ2LdZn
+ML2
+1
+2
+¨
+TdˆRn
+´
+|BtΨ|2 ` κ2T2
+Z2 |∇zΨ|2¯
+dz dx
+`ς ΨLdZnT2
+mL2
+¨
+TdˆRn |U|2σ2σ1 ‹ Ψ dz dx,
+with E0 dimensionless (hence the total energy of the original system is E0 ML2
+T2 ). Therefore, we see
+that the dynamics is encoded by four independent parameters. In what follows, we get rid of a
+parameter by assuming
+ℏT
+mL2 “ 1,
+and we work with the following three independent parameters
+α “ ςΨLdZnT2
+mL2
+mL2
+ℏT ,
+β “ ςZ2
+κ2Ψ
+M
+m,
+c “ κT
+Z .
+It leads to
+ˆ
+iBtU ` 1
+2∆xU
+˙
+pt, xq “ α
+ˆˆ
+TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dy dz
+˙
+Upt, xq,
+(69a)
+´ 1
+c2 B2
+ttΨ ´ ∆zΨ
+¯
+pt, x, zq “ ´βσ2pzq
+ˆˆ
+Td σ1px ´ yq|Upt, yq|2 dy
+˙
+(69b)
+together with
+E0 “ 1
+2
+ˆ
+Td |∇xU|2 dx ` 1
+2
+α
+β
+¨
+TdˆRn
+´ 1
+c2 |BtΨ|2 ` |∇zΨ|2¯
+dz dx
+`α
+¨
+TdˆRn |U|2σ2σ1 ‹ Ψ dz dx.
+This relation allows us to interpret the scaling parameters as weights in the energy balance. Now,
+for notational convenience, we decide to work with a m
+M
+b
+α
+β Ψ instead of Ψ and
+b
+M
+mU instead
+of U; it leads to (3a)-(3c) and (8) with γ “
+b
+M
+m
+?αβ.
+Accordingly, we shall implicitly work
+with solutions with amplitude of magnitude unity.
+The regime where c Ñ 8, with α, β fixed
+leads, at least formally, to the Hartree system (1a)-(1b); arguments are sketched in Appendix B.
+The smallness condition (9) makes a threshold appear on the coefficients in order to guaranty the
+stability: since it involves the product M
+mαβ, it can be interpreted as a condition on the strength of
+the coupling between the particle and the environment, and on the amplitude of the wave function.
+We shall see in the proof that a sharper condition can be derived, expressed by means of the Fourier
+coefficients of the form function σ1.
+54
+
+B
+From Schödinger-Wave to Hartree
+In this Section we wish to justify that solutions – hereafter denoted Uc – of (3a)-(3c) converge to the
+solution of (1a)-(1b) as c Ñ 8. We adapt the ideas in [10] where this question is investigated for
+Vlasov equations. Throughout this section we consider a sequence of initial data UInit
+c
+, ΨInit
+c
+, ΠInit
+c
+such that
+sup
+cą0
+ˆ
+Td |UInit
+c
+|2 dx “ M0 ă 8,
+(70a)
+sup
+cą0
+ˆ
+Td |∇xUInit
+c
+|2 dx “ M1 ă 8,
+(70b)
+sup
+cą0
+" 1
+2c2
+¨
+TdˆRn |ΠInit
+c
+|2 dz dx ` 1
+2
+¨
+TdˆRn |∇zΨInit
+c
+|2 dz dx
+*
+“ M2 ă 8,
+(70c)
+sup
+cą0
+¨
+|UInit
+c
+|2σ1 ‹ σ2|ΨInit
+c
+| dz dx “ M3 ă 8.
+(70d)
+There are several direct consequences of these assumptions:
+• The total energy is initially bounded uniformly with respect to c ą 0,
+• In fact, we shall see that the last assumption can be deduced from the previous ones.
+• Since the L2 norm of Uc is conserved by the equation, we already know that
+Uc is bounded in L8p0, 8; L2pTdqq.
+Next, we reformulate the expression of the potential, separating the contribution due to the
+initial data of the wave equation and the self-consistent part. By using the linearity of the wave
+equation, we can split
+Φc “ ΦInit,c ` ΦCou,c
+where ΦInit,c is defined from the free-wave equation on Rn and initial data ΨInit
+c
+, ΠInit
+c
+:
+1
+c2 B2
+ttΥc ´ ∆zΨ “ 0,
+pΥc, BtΥcq
+ˇˇ
+t“0 “ pΨInit
+c
+, ΠInit
+c
+q.
+(71)
+Namely, we set
+ΦInit,cpt, xq
+“
+ˆ
+Rn σ2pzqσ1 ‹ Υcpt, x, zq dz
+“
+ˆ
+Rn
+´
+cospc|ξ|tqσ1 ‹ pΨInit
+c
+px, |ξq ` sinpc|ξ|t
+c|ξ|
+σ1 ‹ pΨInit
+c
+px, |ξq
+¯pσ2pξq dξ
+p2πqn .
+Accordingly rΨc “ Ψc ´ Υc satisfies
+1
+c2 B2
+tt rΨc ´ ∆z rΨc´ “ ´γσ2σ1 ‹ |Uc|2,
+prΨc, Bt rΨcq
+ˇˇ
+t“0 “ p0, 0q.
+(72)
+55
+
+and we get
+ΦCou,cpt, xq
+“
+γ
+ˆ
+Rn σ2pzqσ1 ‹ rΨcpt, x, zq dz
+“
+γ2c2
+ˆ t
+0
+ˆ
+Rn
+sinpc|ξ|sq
+c|ξ|
+Σ ‹ |Uc|2pt ´ s, xq|pσ2pξq|2
+dξ
+p2πqn ds
+“
+γ2
+ˆ ct
+0
+ˆˆ
+Rn
+sinpτ|ξ|q
+|ξ|
+|pσ2pξq|2
+dξ
+p2πqn
+˙
+looooooooooooooooooomooooooooooooooooooon
+“ppτq
+Σ ‹ |Uc|2pt ´ τ{c, xq dτ,
+where it is known that the kernel p is integrable on r0, 8q [10, Lemma 14].
+Lemma B.1 There exists a constant Mw ą 0 such that
+sup
+c,t,x
+|ΦInit,cpt, xq| ď Mw,
+sup
+c,t,x
+|ΦCou,cpt, xq| ď Mw.
+Proof.
+Combining the Sobolev embedding theorem (mind the condition n ě 3) and the standard
+energy conservation for the free linear wave equation, we obtain
+}Υc}L8p0,8;L2pTd;L2n{pn´2qpRnqqq ď C}∇zΥc}L8p0,8;L2pTdˆRnqq ď C
+a
+2M2.
+Applying Hölder’s inequality, we are thus led to:
+|ΦInit,cpt, xq| ď C}σ2}L2n{pn`2qpRnq}σ1}L2pRdq
+a
+2M2,
+(73)
+which proves the first part of the claim. Incidentally, it also shows that (70d) is a consequence of
+(70a) and (70c). Next, we get
+|ΦCou,cpt, xq| ď γ}Σ}L8pTdq}Uc}L8pr0,8q,L2pTdqq
+ˆ 8
+0
+|ppτq| dτ.
+Corollary B.2 There exists a constant MS ą 0 such that
+sup
+c,t
+}∇Ucpt, ¨q}L2pTdq ď MS.
+Proof.
+This is a consequence of the energy conservation (the total energy being bounded by
+virtue of (70b)-(70d)) where the coupling term
+ˆ
+TdpΦInit,c ` ΦCou,cq|Uc|2 dx
+can be dominated by 2MwM0.
+Coming back to
+BtUc “ ´ 1
+2i∆xUc ` γ
+i pΦInit,c ` ΦCou,cqUc
+(74)
+56
+
+we see that BtUc is bounded in L2p0, 8; H´1pTdqq. Combining the obtained estimates with Aubin-
+Simon’s lemma [44, Corollary 4], we deduce that
+Uc is relatively compact in in C0pr0, Ts; LppTdqq, 1 ď p ă
+2d
+d ´ 2,
+for any 0 ă T ă 8. Therefore, possibly at the price of extracting a subsequence, we can suppose
+that Uc converges strongly to U in C0pr0, Ts; L2pTdqq. It remains to pass to the limit in (74). The
+difficulty consists in letting c go to 8 in the potential term and to justify the following claim.
+Lemma B.3 For any ζ P C8
+c pp0, 8q ˆ Tdq, we have
+lim
+cÑ8
+ˆ 8
+0
+ˆ
+TdpΦInit,c ` ΦCou,cqUcζ dx dt “ γκ
+ˆ 8
+0
+ˆ
+Td Σ ‹ |Uc|2 Ucζ dx dt.
+Proof.
+We expect that ΦCou,c converges to γκΣ ‹ |U|2:
+ˇˇΦCou,cpt, xq ´ γκΣ ‹ |U|2pt, xq
+ˇˇ
+“ γ
+ˇˇˇˇ
+ˆ ct
+0
+Σ ‹ |Uc|2pt ´ τ{c, xqppτq dτ ´ κΣ ‹ |U|2pt, xq
+ˇˇˇˇ
+ď γ
+ˆ ct
+0
+ˇˇˇΣ ‹ |Uc|2pt ´ τ{c, xq ´ Σ ‹ |U|2pt, xq
+ˇˇˇ |ppτq| dτ ` γ
+ˆ 8
+ct
+|ppτq| dτ ˆ }Σ ‹ |U|2}L8pp0,8qˆTdq
+ď γ
+ˆ ct
+0
+Σ ‹
+ˇˇ|Uc|2 ´ |U|2ˇˇpt ´ τ{c, xq |ppτq| dτ
+`γ
+ˆ ct
+0
+Σ ‹
+ˇˇ|U|2pt ´ τ{c, xq ´ |U|2pt, xq
+ˇˇ |ppτq| dτ
+`γ
+ˆ 8
+ct
+|ppτq| dτ }Σ}L8pTdq}U}L8pp0,8q;L2pTdqq.
+Let us denote by Icpt, xq, IIcpt, xq, IIIcptq, the three terms of the right hand side. Since p P L1pr0, 8qq,
+for any t ą 0, IIIcptq tends to 0 as c Ñ 8, and it is dominated by }p}L1pr0,8q}Σ}L8pTdqM0. Next,
+we have
+|Icpt, xq|
+ď
+}p}L1pr0,8q}Σ}L8pTdq sup
+sě0
+ˆ
+Td
+ˇˇ|Uc|2 ´ |U|2ˇˇps, yq dy
+ď
+}p}L1pr0,8q}Σ}L8pTdq sup
+sě0
+ˆˆ
+Td |Uc ´ U|2ps, yq dy ` 2Re
+ˆ
+TdpUc ´ UqUps, yq dy
+˙
+which also goes to 0 as c Ñ 8 and is dominated by 2M0}p}L1pr0,8qq}Σ}L8pTdq. Eventually, we get
+|IIcpt, xq| ď }Σ}L8pTdq
+ˆ ct
+0
+ˆˆ
+Td
+ˇˇ|U|2pt ´ τ{c, yq ´ |U|2pt, yq
+ˇˇ dy
+˙
+|ppτq| dτ.
+Since U P C0pr0, 8q; L2pTdqq, with }Upt, ¨q}L2pTdq ď M0, we can apply the Lebesgue theorem to
+show that IIcpt, xq tends to 0 for any pt, xq fixed, and it is dominated by 2M0}p}L1pr0,8qq}Σ}L8pTdq.
+This allows us to pass to the limit in
+ˆ 8
+0
+ˆ
+Td ΦCou,cUcζ dx dt ´ κ
+ˆ 8
+0
+ˆ
+Td Σ ‹ |U|2Uζ dx dt
+“
+ˆ 8
+0
+ˆ
+Td ΦCou,cpUc ´ Uqζ dx dt `
+ˆ 8
+0
+ˆ
+Td
+´
+ΦCou,c ´ γκΣ ‹ |U|2¯
+Uζ dx dt.
+57
+
+It remains to justify that
+lim
+cÑ8
+ˆ 8
+0
+ˆ
+Td Φinit,cUcζ dx dt “ 0.
+The space variable x is just a parameter for the free wave equation (71), which is equally satisfied
+by σ1 ‹ Υc, with initial data σ1 ‹ pΨInit
+c
+, ΠInit
+c
+q. We appeal to the Strichartz estimate for the wave
+equation, see [26, Corollary 1.3] or [45, Theorem 4.2, for the case n “ 3],which yields
+c1{p
+˜ˆ 8
+0
+ˆˆ
+Rn |σ1 ‹ Υcpt, x, yq|q dy
+˙p{q
+dt
+¸1{p
+ď C
+ˆ 1
+c2
+ˆ
+Rn |σ1 ‹ ΠInit
+c
+px, zq|2 dz `
+ˆ
+Rn |σ1 ‹ ∇yΨInit
+c
+px, zq|2 dz
+˙1{2
+,
+for any admissible pair:
+2 ď p ď q ď 8,
+1
+p ` n
+q “ n
+2 ´ 1,
+2
+p ` n ´ 1
+q
+ď n ´ 1
+2
+,
+pp, q, nq � p2, 8, 3q.
+The L2 norm with respect to the space variable of the right hand side is dominated by
+b
+}σ1}L1pTdq M2.
+It follows that
+ˆ
+Td
+˜ˆ 8
+0
+ˆˆ
+Rn |σ1 ‹ Υcpt, x, zq|q dz
+˙p{q
+dt
+¸2{p
+dx ď C2}σ1}L1pRdq M2
+1
+c2{p ÝÝÝÑ
+cÑ8 0.
+Repeated use of the Hölder inequality (with 1{p ` 1{p1 “ 1) leads to
+ˇˇˇˇ
+ˆ 8
+0
+ˆ
+Td UcζΦInit,c dx dt
+ˇˇˇˇ
+ď
+˜ˆ
+Td
+ˆˆ 8
+0
+|Ucζpt, xq|p1 dt
+˙2{p1
+dx
+¸1{2 ˜ˆ
+Td
+ˆˆ 8
+0
+|ΦInit,cpt, xq|p dt
+˙2{p
+dx
+¸1{2
+.
+On the one hand, assuming that ζ is supported in r0, Rs ˆ Td and p ą 2, we have
+ˆ
+Td
+ˆˆ 8
+0
+|Ucζ|p1 dt
+˙2{p1
+dx
+ď
+ˆ
+Td
+ˆˆ R
+0
+|Uc|2 dt
+˙ ˆˆ R
+0
+|ζ|2p1{p2´p1q dt
+˙p2´p1q{p1
+dx
+ď
+R1`p2´p1q{p1}ζ}L8pp0,8qˆTdq}Uc}L8pp0,8q;L2pTdqq
+which is thus bounded uniformly with respect to c ą 0. On the other hand, we get
+ˆ
+Td
+ˆˆ 8
+0
+|ΦInit,cpt, xq|p dt
+˙2{p
+dx “
+ˆ
+Td
+ˆˆ 8
+0
+ˇˇˇ
+ˆ
+Rn σ2pzqσ1 ‹ Υcpt, x, zq dz
+ˇˇˇ
+p
+dt
+˙2{p
+dx
+ď }σ2}Lq1pRnq
+ˆ
+Td
+ˆˆ 8
+0
+ˇˇˇ
+ˆ
+Rn |σ1 ‹ Υcpt, x, zq|q dz
+ˇˇˇ
+p{q
+dt
+˙2{p
+dx
+which is of the order Opc´2{pq.
+58
+
+C
+Well-posedness of the Schrödinger-Wave system
+The well-posedness of the Schrödinger-Wave system is justified by means of a fixed point argument.
+The method described here works as well for the problem set on Rd, and it is simpler than the
+approach in [21] since it avoids the use of “dual” Strichartz estimates for the Schrödinger and the
+wave equations.
+We define a mapping that associates to a function pt, xq P r0, Ts ˆ Td ÞÑ V pt, xq P C:
+• first, the solution Ψ of the linear wave equation
+1
+c2 B2
+ttΨ ´ ∆zΨ “ ´σ2σ1 ‹ |V |2,
+pΨ, BtΨq
+ˇˇ
+t“0 “ pΨ0, Ψ1q;
+• next, the potential Φ “ σ1 ‹
+´
+Rn σ2Ψ dz;
+• and finally the solution of the linear Schrödinger equation
+iBtU ` 1
+2∆xU “ γΦU,
+U
+ˇˇ
+t“0 “ UInit.
+These successive steps define a mapping S : V ÞÝÑ U and we wish to show that this mapping
+admits a fixed point in C0pr0, Ts; L2pTdqq, which, in turn, provides a solution to the non linear
+system (3a)-(3c). In this discussion, the initial data UInit, Ψ0, Ψ1 are fixed once for all in the space
+of finite energy:
+UInit P H1pTdq,
+Ψ0 P L2pTd; .H1pRnqq,
+Ψ1 P L2pTd ˆ Rnq.
+We observe that
+d
+dt
+ˆ
+Td |U|2 dx “ 0.
+Hence, the mapping S applies the ball Bp0, }UInit}L2pTdqq of C0pr0, Ts; L2pTdqq in itself; we thus
+consider U “ SpV q for V P C0pr0, Ts; L2pTdqq such that }V pt, ¨q}L2pTdq ď }UInit}L2pTdq. Moreover,
+we can split
+Ψ “ Υ ` rΨ
+with Υ solution of the free wave equation
+1
+c2 B2
+ttΥ ´ ∆zΥ “ 0,
+pΥ, BtΥq
+ˇˇ
+t“0 “ pΨ0, Ψ1q,
+and
+1
+c2 B2
+tt rΨ ´ ∆z rΨ “ 0,
+pΥ, Bt rΨq
+ˇˇ
+t“0 “ 0.
+We write Φ “ ΦI ` rΦ for the associated splitting of the potential. In particular, the standard
+energy conservation for the wave equation tells us that
+1
+2c2
+¨
+TdˆRn |BtΥ|2 dz dx ` 1
+2
+¨
+TdˆRn |∇zΥ|2 dz dx
+“
+1
+2c2
+¨
+TdˆRn |Ψ1|2 dz dx ` 1
+2
+¨
+TdˆRn |∇zΨ0|2 dz dx “ M2
+59
+
+holds. It follows that
+|ΦIpt, xq| ď C}σ2}L2n{pn`2pRnq}σ1}L2pTdq
+a
+2M2
+by using Sobolev’s embedding. Next, we obtain
+rΦpt, xq
+“
+ˆ
+Rn σ2pzqσ1 ‹ rΨpt, x, zq dz
+“
+γ
+ˆ ct
+0
+ˆˆ
+Rn
+sinpτ|ξ|q
+|ξ|
+|pσ2pξq|2
+dξ
+p2πqn
+˙
+looooooooooooooooooomooooooooooooooooooon
+“ppτq
+Σ ‹ |V |2pt ´ τ{c, xq dτ,
+which thus satisfies
+sup
+xPTd |rΦpt, xq| ď γ}Σ}L8pTdq
+ˆ ct
+0
+|ppτq|
+ˆˆ
+Td |V |2pt ´ τ{c, yq dy
+˙
+dτ.
+In particular
+|rΦpt, xq| ď γ}Σ}L8pTdq}p}L1pp0,8qq}V }C0pr0,Ts;L2pTdqq ď γ}Σ}L8pTdq}p}L1pp0,8qq}UInit}L2pTdq
+lies in L8pp0, Tq ˆ Tdq, and thus Φ P L8pp0, Tq ˆ Rdq. This observation guarantees that U “ ;SpV q
+is well-defined.
+Thus, let us pick V1, V2 in this ball of C0pr0, Ts; L2pTdqq and consider Uj “ SpVjq. We have
+iBtpU2 ´ U1q ` 1
+2∆xpU2 ´ U1q “ γΦ2pU2 ´ U1q ` γpΦ2 ´ Φ1qU1,
+pU2 ´ U1q
+ˇˇ
+t“0 “ 0.
+It follows that
+d
+dt
+ˆ
+Td |U2 ´ U1|2 dx “ 2γIm
+ˆˆ
+TdpΦ2 ´ Φ1qU 1pU2 ´ U1q dx
+˙
+ď 2γ}U1}L2pTdq }U2 ´ U1}L2pTdq }Φ2 ´ Φ1}L8pTdq “ 2γ}U1}L2pTdq }U2 ´ U1}L2pTdq }rΦ2 ´ rΦ1}L8pTdq
+ď 2γ2}Σ}L8pTdq}UInit}L2pTdq }U2 ´ U1}L2pTdq
+ˆ ct
+0
+|ppτq|
+ˆˆ
+Td
+ˇˇ|V2|2 ´ |V1|2ˇˇpt ´ τ{c, yq dy
+˙
+dτ.
+We use the elementary estimate
+ˆ
+Td
+ˇˇ|V2|2´|V1|2ˇˇ dy “
+ˆ
+Td
+ˇˇ|V2´V1|2`2RepV2´V1qV1
+ˇˇ dy ď }V2´V1}2
+L2pTdq`2}V2´V1}L2pTdq }V1}L2pTdq.
+Combining this with Cauchy-Schwarz and Young inequalities, we arrive at
+d
+dt
+ˆ
+Td |U2 ´ U1|2 dx
+ď 2γ2}Σ}L8pTdq}UInit}L2pTdq
+ˆ
+2}UInit}L2pTdq
+ˆ ct
+0
+|ppτq|}V2 ´ V1}2pt ´ τ{cqL2pTdq dτ
+`}U2 ´ U1}L2pTdq2}UInit}L2pTdq
+ˆ ct
+0
+|ppτq|}V2 ´ V1}pt ´ τ{cqL2pTdq dτ
+˙
+ď 2γ2}Σ}L8pTdq}UInit}2
+L2pTdq
+´
+}U2 ´ U1}2
+L2pTdq
+`p2 ` }p}L1pp0.8qq
+ˆ ct
+0
+|ppτq|}V2 ´ V1}2pt ´ τ{cqL2pTdq dτ
+˙
+.
+60
+
+Set L “ 2γ2}Σ}L8pTdq}UInit}2
+L2pTdq. We deduce that
+}U2 ´ U1}ptq2
+L2pTdq ď p2 ` }p}L1pp0.8qqL
+ˆ t
+0
+eLpt´sq
+ˆ cs
+0
+|ppτq|}V2 ´ V1}2ps ´ τ{cqL2pTdq dτ ds.
+We use this estimate for 0 ď t ď T ă 8 and we obtain
+}U2 ´ U1}ptq2
+L2pTdq ď p4 ` }p}L1pp0.8qqLTeLT }p}L1pp0.8q sup
+0ďsďT
+}V2 ´ V1}2psqL2pTdq.
+Hence for T small enough, S is a contraction in C0pr0, Ts; L2pTdqq, and consequently it admits a
+unique fixed point. Since the fixed point still has its L2 norm equal to }UInit}L2pTdq, the solution
+can be extended on the whole interval r0, 8q. The argument can be adapted to handle the Hartree
+system.
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