Results on the spectral stability of standing wave solutions of the Soler model in 1-D
Danko Aldunate, Julien Ricaud, Edgardo Stockmeyer, Hanne Van Den Bosch
aa r X i v : . [ m a t h - ph ] F e b RESULTS ON THE SPECTRAL STABILITY OF STANDING WAVESOLUTIONS OF THE SOLER MODEL IN 1-D
DANKO ALDUNATE, JULIEN RICAUD, EDGARDO STOCKMEYER,AND HANNE VAN DEN BOSCH
Abstract.
We study the spectral stability of the nonlinear Dirac operator in dimen-sion 1+1, restricting our attention to nonlinearities of the form f ( h ψ, σ ψ i C ) σ . Weobtain bounds on eigenvalues for the linearized operator around standing wave solu-tions of the form e − iωt φ . For the case of power nonlinearities f ( s ) = | s | p , p > ω such that the linearized operator has no unstableeigenvalues on the axes of the complex plane. As a crucial part of the proofs, weobtain a detailed description of the spectra of the self-adjoint blocks in the linearizedoperator. In particular, we show that the condition h φ , σ φ i C > groundstates analogously to the Schr¨odinger case. Contents
1. Introduction 21.1. Main results 51.2. Outline of the paper 72. Preliminaries 83. Bifurcations from the origin 134. Lower bound on Re z β ( p ) 266. The massive Gross–Neveu model 297. L has only one eigenvalue in ( − ω,
0) 317.1. Strategy of the proof of Theorem 1.8 317.2. Spectrum of L µ in the non-relativistic limit 33Appendix 38References 40 Date : February 24, 2021. Introduction
We consider the nonlinear Dirac equation in 1 + 1 dimensions with Soler-type nonlin-earity f : R → R and initial data φ ∈ H ( R , C ), given by ( i∂ t ψ = D m ψ − f ( h ψ, σ ψ i C ) σ ψ ,ψ ( · ,
0) = φ , (1)where ψ = ( ψ , ψ ) T : R × R → C and D m = iσ ∂ x + mσ is the one-dimensional Dirac operator with mass m >
0. Here, the σ j ’s are the standardPauli matrices σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) and σ = (cid:18) − (cid:19) . Other conventions amount to permuting the roles of σ , σ and σ in this equation. Theadvantage of the present form is that no complex coefficients appear.This type of models describes the dynamics of spinors self-interacting through a mass-like potential. They were introduced in [20] and further developed in [29]. The case f ( s ) = s corresponds to the classical massive Gross–Neveu model [18].We assume that the nonlinearity f satisfies Assumption 1.1. f ∈ C ( R \ { } , R ) ∩ C ( R , R ), f (0) = 0, lim s →∞ f ( s ) = + ∞ .Moreover, f ′ ( s ) > s >
0, and lim s → + sf ′ ( s ) = 0.Under these conditions, we know from [8] (see also [6, Chapter IX]) that equation (1)has standing wave solutions of the form ψ ( x, t ) = e − iωt φ ( x ) (2)for all ω ∈ (0 , m ), where the initial condition φ solves( D m − ω ) φ − f ( h φ , σ φ i C ) σ φ = 0 . (3)Moreover, φ ≡ ( v, u ) T ∈ H ( R ) is continuous, decays exponentially at rate √ m − ω ,can be chosen to be real-valued, and such that v is even, u is odd. In the following, wealways assume that φ is the unique solution to (3) such that v (0) > u (0) = 0. Then,we also have h φ , σ φ i C = v − u > . (4)Notice that φ ≡ φ ( ω ) depends on ω ∈ (0 , m ).In this paper we study the spectral stability of these solitary waves and obtain newspectral properties of the linearized operator of equation (1) around a solution of theform (2). In order to set up the problem, let us define the family of operators in L ( R , C ) L µ ≡ L µ ( ω ) := D m − ω − f ( h φ , σ φ i C ) σ − µQ, with domain H ( R , C ) , (5)parametrized by µ ∈ R , where Q acts as the matrix-valued multiplication operator Q := f ′ ( h φ , σ φ i C ) ( σ φ ) ( σ φ ) T . (6) Remark 1.2.
Since φ depends on ω , Q depends on ω as well. Occasionally, whenneeded for clarity, we will highlight this dependency by writing Q ω . N SPECTRAL STABILITY OF SOLER STANDING WAVES 3
As we will see, L and L are the operators appearing in the linearization of thenonlinear equation (1). However, the key idea in our study of spectral stability is toconsider the analytic operator family µ L µ instead.Looking for solutions to (1) of the form ψ ( x, t ) = e iωt ( φ ( x ) + ρ ( x, t )), the formallinearization of (1) around the solitary wave solution φ takes the form i∂ t ρ = L ρ − f ′ ( h φ , σ φ i C ) Re ( h φ , σ ρ i C ) σ φ . Using that h α, β i α = αα T β for any α, β ∈ C , and that φ is real-valued, this equationcan be written as i∂ t ρ = iL Im ρ + L Re ρ. Or, equivalently, i∂ t (cid:18) Re ρi Im ρ (cid:19) = (cid:18) L L (cid:19) (cid:18) Re ρi Im ρ (cid:19) := H (cid:18) Re ρi Im ρ (cid:19) . Here H is the linearized operator. For µ ∈ C , we define the family of closed operators H µ ≡ H µ ( ω ) := (cid:18) L L µ (cid:19) , with domain H ( R , C ) . (7)Note that we take the convention to keep the complex number i at the left hand sideof the linearized equation, which is not the usual convention in the PDE literature. Asa consequence, H is self-adjoint, and H µ has essential spectrum on the real axis for all µ ∈ C . With this convention, spectral stability corresponds to the absence of eigenvaluesof H with positive imaginary part (compare, for instance, with the definition givenin [4]). Since, as we will see, the spectrum of H is symmetric with respect to the realaxis, spectral stability amounts to all eigenvalues of H being real.In the case of the nonlinear Schr¨odinger and Klein–Gordon operators, the spectraland orbital stability of solitary waves is well-understood since mayor breakthroughsin the ‘80 [17, 32]. However, this is not the case for the Dirac analogues. The maintechnical difficulties are related to the lack of positivity of the Dirac operator. A notableexception is [26], where the authors prove orbital stability in the one dimensional massiveThirring model, which is completely integrable and the result relies on a coercive higherorder conserved quantity.In the Schr¨odinger case, spectral stability is crucial to characterize the orbital stabilityand, together with some additional assumptions, also implies asymptotic stability (seefor instance [12]). For the Dirac equation, this connection is not completely clear.Nevertheless, in dimension three, asymptotic stability of small amplitude solitary waves(that is, ω close to m ) is shown in [7] to follow from spectral stability and severaltechnical assumptions.Spectral stability in the Soler model with nonlinearity f ( s ) = | s | p for dimensions 1, 2and 3 is studied in [5, 10]. In these works, results are obtained for ω sufficiently close to m , using the convergence to the corresponding nonlinear Schr¨odinger equation. In [5],it is shown that there exists an interval for spectral stability for powers 1 < p p > p = 1 (massive Gross–Neveu model) has been studied D. ALDUNATE, J. RICAUD, E. STOCKMEYER, AND H. VAN DEN BOSCH numerically in [1]. The general conjecture seems to be that the model is spectrally stable,although there has been some controversy [23] for the case of small frequencies ω .Still in one dimension, but when translation invariance is broken by a potential addedto the Dirac operator, spectral and asymptotic stability for large ω is shown in [27]. Onthe other hand, when translation invariance is broken by switching off the nonlinearityaway from the origin, a recent paper [3] shows spectral stability and instability byexplicit computations. Interestingly, as long as the nonlinearity preserves parity , alleigenvalues are real or purely imaginary.Before describing our results, we recall some well-known properties of the operators H µ and L µ , see e.g. [6]. For the convenience of the reader, we give a proof of those inSection 2. Proposition 1.3.
Let f satisfy Assumption 1.1, µ ∈ R and consider L µ and H µ definedin (5) and (7) . Then,(i) σ ess ( L µ ) = ( −∞ , − m − ω ] ∪ [ m − ω, + ∞ ) .(ii) − ω and are simple eigenvalues of L with eigenfunctions φ − ω := σ φ and φ ,respectively. The spectrum of L is symmetric with respect to − ω .(iii) − ω and are simple eigenvalues of L with eigenfunctions φ − ω and ∂ x φ ,respectively.(iv) The spectrum of H µ is symmetric with respect to the real and imaginary axis.That is, if z is an eigenvalue of H µ , then ± z and − z are also eigenvalues.(v) σ ess ( H µ ) = ( −∞ , − m + ω ] ∪ [ m − ω, + ∞ ) .(vi) ± ω are eigenvalues of H and is a double eigenvalue. By standard Kato perturbation theory [21], away from the essential spectrum, eigen-values of H µ are branches of analytic functions in the parameter µ . Since H is self-adjoint, it has only real eigenvalues. Our point of view is to check whether eigenvaluesof H µ can become complex when µ increases from 0 to 2.For the case of Schr¨odinger operators, the positivity of one of the block-operatorsimmediately implies that all eigenvalues are either real or purely imaginary. Therefore,eigenvalues can leave the real axis only by passing through zero, and this passing throughzero is characterized by the Vakhitov–Kolokolov criterion [22, 31].Our first result gives a completely analogous characterization of eigenvalue branchespassing through zero when µ increases. Unfortunately, in the Dirac case, this is notenough to prove spectral stability: we are not aware of any argument to exclude eigen-value branches away from the axes of the complex plane. Proposition 1.3(iv) showsthat eigenvalues in the gap of the essential spectrum ( − m + ω, m − ω ) can only leavethe real axis if they are degenerate. Furthermore, as described for instance in [4],where the point of view are perturbations in ω , pairs of non-real eigenvalues couldoriginate from the thresholds of the essential spectrum at ± ( m + ω ) and ± ( m − ω ),and also appear from embedded eigenvalues between the thresholds. As shown in [4],embedded eigenvalues can only exist between the inner and outer thresholds, i.e, in[ − m − ω, − m + ω ] ∪ [ m − ω, m + ω ]. On the other hand, we are not aware of examples where eigenvalues occur off the axes of thecomplex plane.
N SPECTRAL STABILITY OF SOLER STANDING WAVES 5
For a more complete account on the spectral stability of Dirac equations, we referthe reader to the recent monograph [6] by Boussa¨ıd and Comech, and the referencestherein.While eigenvalues away from the axes are not excluded a priori, we will see thateigenfunctions associated to these eigenvalues satisfy the orthogonality condition (21)in Section 4. This allows us to give better estimates for these eigenvalues and, forthe case of sufficiently large ω , these bounds imply that non-real eigenvalues can onlyappear close to the embedded thresholds ± ( m + ω ).1.1. Main results.
Our first result justifies the approach of studying the operatorfamily µ H µ . Theorem 1.4.
Let f satisfy Assumption 1.1 and assume further that ω ∈ (0 , m ) and f are such thati) L has a single eigenvalue in ( − ω, ,ii) ∂ ω || φ ( ω ) || L .Let µ ∈ (0 , and H µ be the associated linearized operator defined in (7) . The algebraicmultiplicity of zero, as an eigenvalue of H µ , equals . We recall that the algebraic multiplicity of an eigenvalue is the dimension of thegeneralized null space, see Section 3 below for details.Condition ii) is known as the
Vakhitov–Kolokolov criterion for spectral stability.In [2], the authors study the meaning of the equality case in the Dirac context. Theyshow, when equality holds, that eigenvalue branches can pass through zero as ω varies.Conditions i) and ii) are precisely the generalizations to the Dirac case of the classicalhypotheses in [17, 32]. The results there apply to groundstates of the correspondingSchr¨odinger operator. For the Dirac operator, in view of our next result, we mayinterpret φ − ω and φ as groundstates of L + ω . Theorem 1.5.
Let f satisfy Assumption 1.1 and L defined in (5) . Then, L has noeigenvalues in ( − ω, . Our next results concern bounds for eigenvalues away from the axes of the complexplane. Together with Theorem 1.4, through a continuity argument, they tell us thatwhen µ increases from 0 to 2, no eigenvalues of H µ can reach the imaginary axis withoutpassing through zero. This allows to exclude non-zero purely imaginary eigenvaluesfor H .If the operator norm of Q , denoted by ||| Q ||| ≡ ||| Q ω ||| , is sufficiently small, we obtainbounds for eigenvalues of H µ that are not on the axes. Theorem 1.6.
Let f satisfy Assumption 1.1 and µ > . For any E ∈ [0 , m ) , thereexists ω E ∈ [ E/ , m ) such that, for all ω ∈ [ ω E , m ) and µ ∈ (0 , , if z ∈ C \ ( R ∪ i R ) is an eigenvalue of H µ ( ω ) , then Re z > E . (8) Moreover, without any restriction on ω ∈ (0 , m ) and for any eigenvalue z ∈ C of H µ , | Im z | µ ||| Q ||| . (9) D. ALDUNATE, J. RICAUD, E. STOCKMEYER, AND H. VAN DEN BOSCH
Note that for ω > E/ E m + ω . Thus, the complex values allowed by Theorem 1.6for z always include a neighborhood of the outer thresholds ± ( m + ω ).In order to show the absence of non-zero purely imaginary eigenvalues of H , we useTheorem 1.4 combined with Theorem 1.6 for E = 0 (see Corollary 3.2 below).For power nonlinearities f ( s ) = | s | p , we verify the corresponding conditions for p > ω ’s characterized by a function β ( p ), displayed in (37), taking valuesin (0 , Theorem 1.7.
Let f ( s ) = | s | p , p > , and β ( p ) defined in (37) . Then, H has nonon-zero eigenvalues on the imaginary axis for • p and ω ∈ [ β ( p ) m, m ) ; • p > and ω ∈ h β ( p ) m, p p − m i . These conditions are plotted in Figure 1. In the massive Gross–Neveu model [18], p = 1, we obtain a larger range of ω ’s: one can take β (1) = q √ − ≈ . p ∈ (0 ,
15) Zoom onto p ∈ (0 , . Figure 1.
The blue region in the upper part of the graphs are the pa-rameters ( p, ω/m ) for which we show that Theorem 1.6 holds with E = 0.Note that for p = 1, we obtain an improved range. The green regionin the lower part represents the parameters for which we show that theVakhitov–Kolokolov criterion holds. In the intersection of these regions,Theorem 1.7 holds and, thus, H has no non-zero purely imaginary eigen-values.Let us briefly highlight the novel elements of our results when comparing to thespectral stability results in the limit m → ω in [4]. Unlike the case of [4], we are notable to conclude spectral stability for any values of ( p, ω ). However, for pure powernonlinearities, we obtain an explicit range of parameters that do not admit unstableeigenvalues on the imaginary axis. We obtain results for p ∈ (0 , p >
2, which might be surprising in view of the spectral instability that holds forthese p ’s in the non-relativistic limit [10]. To the best of our knowledge, our results arethe first quantitative results towards spectral stability.The obtained range of parameters is determined by three conditions. On one handthe Vakhitov–Kolokolov criterion (15) and the bounds in Theorem 1.6. These two N SPECTRAL STABILITY OF SOLER STANDING WAVES 7 conditions can be checked once some information on φ is available. The third conditionis Hypothesis i) in Theorem 1.4 on the eigenvalues of L . We prove that this lattercondition holds for all power nonlinearities. Theorem 1.8.
Let f ( s ) = | s | p , p > , and ω ∈ (0 , m ) . Then, L has exactly oneeigenvalue in ( − ω, . We expect this result to hold for groundstates with general f , at least if f behavesas a power for small s .The key idea in the proof of Theorem 1.8 is to show first that the number of eigenvaluesof L ( ω ) in ( − ω,
0) is independent of ω ∈ (0 , m ). Therefore, it is sufficient to check thiscondition in the non-relativistic limit. Since L is a self-adjoint operator, its eigenvaluescan be characterized variationally [13, 14, 28] and we use this property to estimate thenumber of eigenvalues in the limit, see Section 7 for details.1.2. Outline of the paper.
In Section 2, we group some basic properties valid forgeneral f , in particular those summarized in Proposition 1.3. Furthermore, we show inLemma 2.3 that eigenvalues for L µ are simple. The simplicity is important because itensures that no eigenvalue branches can cross as µ or ω vary. We also prove Theorem 1.5by an oscillation argument.In the next section, we recall some facts about analytic perturbation theory for closedoperators and prove Theorem 1.4. Moreover, in Corollary 3.2, we describe how thistheorem and the nonnegativity of Re z yield absence of non-zero purely imaginaryeigenvalues of H .In Section 4, we obtain bounds on Re z for eigenvalues z of H µ . The starting point isan identity on Re z , in Lemma 4.1. This identity allows us to prove Lemma 4.2 whichis, to the best of our knowledge, the first bound on the spectrum of H for generalnonlinearity f . The remainder of the section contains the proof of Theorem 1.6.We then specify to the case of power nonlinearities in Section 5. In this case, thesolutions φ are known explicitly, and this allows us to compute ||| Q ||| and check forwhich range of parameters the Vakhitov–Kolokolov criterion is verified. Next, we insertthe expression for ||| Q ||| in the bounds obtained in Section 4 and conclude the proof ofTheorem 1.7 except for the case p = 1.The case p = 1, or f ( s ) = s , is known as the massive Gross–Neveu model and weobtain some further results for this case in Section 6. Using the explicit resonancesfor L in this model, we show that the only eigenvalues of L are − ω and 0. Pluggingthis into the bounds from Section 4, we show the absence of non-zero purely imaginaryeigenvalues of H for all ω ∈ [0 . m, m ).Finally, the proof of Theorem 1.8 is deferred to section 7, where we start by recall-ing the minmax principle for eigenvalues in a gap and obtain the leading term in theexpansion of eigenvalues of L µ for ω close to m . A few explicit computations that areonly necessary for the expression of the error terms are left for the Appendix. Acknowledgments.
We thank S´ebastien Breteaux, J´er´emy Mougel and Phan Th`anhNam for helpful discussions. The research visits leading to this work where partiallyfunded by ANID (Chile) project REDI–170157. J.R. received funding from the DeutscheForschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excel-lence Strategy (EXC-2111-390814868). E.S. and H. VDB. have been partially funded
D. ALDUNATE, J. RICAUD, E. STOCKMEYER, AND H. VAN DEN BOSCH by ANID (Chile) through Fondecyt project
Preliminaries
The main goal of this section is to prove Theorem 1.5 about the spectrum of L .We first revisit well-known properties of the spectra of the linearized operators (seee.g. [4, 5, 6]). Proposition 1.3 follows from Propositions 2.1 and 2.2 below.Throughout this section, we assume that f satisfies Assumption 1.1. Proposition 2.1.
For all µ ∈ R , L µ is self-adjoint and a bounded relatively compactperturbation of D m − ω . Therefore, σ ess ( L µ ) = ( −∞ , − m − ω ] ∪ [ m − ω, + ∞ ) , and the discrete spectrum of L µ consists of eigenvalues of finite multiplicity in the gap( − m − ω, m − ω ) .Moreover, H µ is a bounded relatively compact perturbation of D := (cid:18) D m − ω D m − ω (cid:19) . In particular, its essential spectrum is σ ess ( H µ ) = ( −∞ , − m + ω ] ∪ [ m − ω, + ∞ ) . The proof of this proposition uses all the assumptions on f made in Assumption 1.1. Proof.
As shown for instance in [1, 6, 8], under Assumption 1.1, the components v, u of φ satisfy ω ( v + u ) − m ( v − u ) + 12 F ( v − u ) = 0 , (10)where F ( s ) := R s f ( r ) d r on R . The assumptions f (0) = 0 and f ′ > , + ∞ ) implythat F > , + ∞ ), and consequently that0 ω ( v + u ) m ( v − u ) . Thus, again since f ′ > , + ∞ ), || Q || C → C ( x ) = ( v + u ) f ′ ( v − u ) mω ( v − u ) f ′ ( v − u ) . (11)Therefore, the property lim s → + sf ′ ( s ) = 0 in Assumption 1.1, together with v − u > | x |→∞ || Q || C → C ( x ) mω lim | x |→∞ ( v ( x ) − u ( x )) f ′ ( v ( x ) − u ( x )) = mω lim s → + sf ′ ( s ) = 0 . Moreover, since v and u are bounded and vanish at infinity, as a consequence of As-sumption 1.1, we have that f ( v ( x ) − u ( x )) σ + µQ ( x )is a bounded symmetric matrix, and vanishes when | x | goes to infinity. Therefore, L µ = D m − ω − f ( v − u ) σ − µQ N SPECTRAL STABILITY OF SOLER STANDING WAVES 9 is self-adjoint and f ( v − u ) σ + µQ is relatively compact with respect to the free Diracoperator (see e.g. [30, Theorem 4.7]). We conclude that σ ess ( L µ ) = σ ( D m − ω ) = ( −∞ , − m − ω ] ∪ [ m − ω, + ∞ ) . The same argument applies to H µ and gives σ ess ( H µ ) = σ ess (D) = σ ess ( D m − ω ) ∪ σ ess ( − D m + ω ) . (cid:3) Proposition 2.2.
Let µ ∈ R .(i) The spectrum of L is symmetric with respect to − ω .(ii) − ω and are eigenvalues of L with eigenfunctions φ − ω := σ φ and φ .(iii) Qφ − ω = 0 and − ω is an eigenvalue of L µ with eigenfunction φ − ω .(iv) − ω and are eigenvalues of L with eigenfunctions φ − ω and ∂ x φ . As aconsequence, and ± ω are eigenvalues of H , with respective eigenfunctions (cid:18) φ (cid:19) , (cid:18) ∂ x φ (cid:19) and (cid:18) φ − ω ∓ φ − ω (cid:19) . (v) The spectrum of H µ is symmetric with respect to the real and imaginary axis.That is, if z is an eigenvalue of H µ , then ± z and − z are also eigenvalues.Proof. The equation L φ = 0 is just a rewriting of the nonlinear ODE (3). Next, weuse the anticommutators { L + ω , σ } = { D m , σ } − f ( h φ , σ φ i C ) { σ , σ } = 0to conclude that σ φ is an eigenfunction associated to − ω , which concludes (ii) , andthat the spectrum of L + ω is symmetric, which is (i) .The definition of Q in (6) gives (iii) since, by properties of Pauli matrices and sincethe components of φ are real, we have( σ φ ) T φ − ω = h σ φ , σ φ i C = 0 . For (iv) , we compute H (cid:18) φ (cid:19) = (cid:18) L φ (cid:19) = 0 and H (cid:18) φ − ω ∓ φ − ω (cid:19) = (cid:18) ∓ L φ − ω L φ − ω (cid:19) = ± ω (cid:18) φ − ω ∓ φ − ω (cid:19) , where we used (iii) in the latter. The second eigenfunction associated to 0 is a conse-quence of φ being real-valued and the definition of Q in (6). Indeed, note thatRe( h ∂ x φ , σ φ i C ) = h ∂ x φ , σ φ i C = ( σ φ ) T ∂ x φ . So we obtain L ∂ x φ = ∂ x ( L φ ) + 2 f ′ ( h φ , σ φ i C ) Re( h ∂ x φ , σ φ i C ) σ φ = 2 Q∂ x φ , and conclude that ∂ x φ is an eigenfunction of L associated to the eigenvalue 0, i.e., L ∂ x φ = ( L − Q ) ∂ x φ = 0 . By definition of H , we have H (cid:18) ∂ x φ (cid:19) = (cid:18) L ∂ x φ (cid:19) = 0 . The last statement of this proposition is an immediate consequence of the fact that H µ is equal to its complex conjugate operator and that S H µ = − H µ S , where S = σ ⊗ × = (cid:18) × − × (cid:19) . (cid:3) Notice that the operator L + ω may be interpreted as a Dirac operator with theeffective mass M := m − f (cid:0) v − u (cid:1) . (12)This Dirac operator is connected to the Schr¨odinger operators − ∂ x + M ∓ M ′ . We usethis connection to prove the groundstate theorem below. Proof of Theorem 1.5.
We already know (Proposition 2.2) that {− ω, } ⊂ σ ( L ), sowe have to prove that there is no eigenvalues in ( − ω ; 0). Defining A := L + ω , weneed to show that A has no spectrum in ( − ω, + ω ).For this, it is convenient to use the change of basis U = 1 √ (cid:18) − (cid:19) = 1 √ σ + σ )to obtain the unitary equivalence, using the properties of the Pauli matrices, U AU = − i∂ x σ + M σ = (cid:18) − ∂ x + M∂ x + M (cid:19) , where M is defined in (12). From this, we find that A is unitarily equivalent to( U AU ) = (cid:18) ( − ∂ x + M )( ∂ x + M ) 00 ( ∂ x + M )( − ∂ x + M ) (cid:19) = (cid:18) − ∂ x + M − M ′ − ∂ x + M + M ′ (cid:19) , which is a diagonal matrix with two Schr¨odinger operators (with essential spectrum[ m , + ∞ )) on the diagonal. We compute( U AU ) U φ = U A φ = U (cid:0) L + 2 ωL + ω (cid:1) φ = ω U φ and √ U φ = (cid:18) v + uv − u (cid:19) . Since, ( v + u )( v − u ) = v − u > v ± u ∈ C ( R )do not change sign and are eigenfunctions of − ∂ x + M ∓ M ′ , respectively, associatedto the same eigenvalue ω . Therefore, they are the respective groundstates by Sturm’soscillation theorem, and A has no eigenvalue below ω . We conclude that A := L + ω has no eigenvalues in ( − ω, + ω ). (cid:3) We continue this section by proving the simplicity of eigenvalues of L µ . Lemma 2.3.
Let µ ∈ R . The eigenvalues of L µ are simple. N SPECTRAL STABILITY OF SOLER STANDING WAVES 11
Proof.
Let λ ∈ R be an eigenvalue of L µ and assume φ = ( f , g ) T and φ = ( f , g ) T to be eigenfunctions associated to λ . The equation L µ φ j = λφ j can be written as ∂ x φ j = − iλσ φ j − M ( x ) σ φ j − µσ Q ( x ) φ j , with M defined in (12). Furthermore, we can decompose µQ ( x ) = q ( x ) C + q ( x ) σ + q ( x ) σ , for some functions q , q and q whose explicit expressions are not necessary to completethe proof. Using the identity σ m σ k = i X l =1 ǫ mkl σ l where ǫ mkl is the completely antisymmetrix tensor such that ǫ = 1, we finally rewritethe eigenvalue equation as ∂ x φ j = ( − iλ + µq ) σ φ j − ( M ( x ) + q ( x )) σ φ j − µq ( x ) σ φ j . Now, define the determinant W ( x ) := det (cid:0) φ ( x ) | φ ( x ) (cid:1) and compute W ′ = det (cid:0) φ ′ | φ (cid:1) + det (cid:0) φ | φ ′ (cid:1) = ( − iλ + µq ) (cid:0) det (cid:0) σ φ | φ (cid:1) + det (cid:0) φ | σ φ (cid:1)(cid:1) + ( M ( x ) + q ( x )) (cid:0) det (cid:0) σ φ | φ (cid:1) + det (cid:0) φ | σ φ (cid:1)(cid:1) − µq ( x ) (cid:0) det (cid:0) σ φ | φ (cid:1) + det (cid:0) φ | σ φ (cid:1)(cid:1) . For k ∈ { , , } , we computedet (cid:0) σ k φ | φ (cid:1) = det (cid:0) σ k φ | σ k φ (cid:1) = det( σ k ) det (cid:0) φ | σ k φ (cid:1) = − det (cid:0) φ | σ k φ (cid:1) , so we conclude W ′ ≡
0. Because φ i ∈ L ( R ), this implies that W ≡
0. Then, for each x ∈ R , φ ( x ) and φ ( x ) are colinear vectors in C . If φ (0) = 0, there exists α ∈ C such that φ (0) = αφ (0). By linearity and uniqueness of the solution to the Cauchyproblem (cid:26) L µ Ψ = λ ΨΨ(0) = φ (0) = α (0) φ (0) , this implies that φ ( x ) = αφ ( x ) for all x ∈ R . If φ (0) = 0, the same uniqueness resultimplies that φ ≡
0, contradicting that φ is an eigenfunction. (cid:3) We conclude this section with the following lemma, which is (9) and the first step inthe proof of (8), both in Theorem 1.6.
Lemma 2.4.
Let f satisfy Assumption 1.1 and µ > . If z ∈ C is an eigenvalue of H µ ,then | Im z | µ ||| Q ||| . Moreover, lim ω → m ||| Q ω ||| = 0 . Proof.
For the first point, denoting ϕ = ( ϕ , ϕ ) T the eigenvector associated to z , theeigenvalue equation reads ( L ϕ = zϕ ,L µ ϕ = zϕ . (13a)(13b)Summing the inner product of (13a) and (13b), respectively with ϕ and ϕ , gives h ϕ , L ϕ i + h ϕ , L ϕ i − h ϕ , µQϕ i = z || ψ || . Since L is self-adjoint, h ϕ , L ϕ i + h ϕ , L ϕ i ∈ R and we haveIm z || ψ || = µ Im h Qϕ , ϕ i . By Cauchy–Schwarz inequality, it yields | Im z | µ ||| Q ||| || ϕ || || ϕ |||| ϕ || + || ϕ || µ ||| Q ||| . For the second point, we start by computing v (0). Evaluating (10) at x = 0 gives2( m − ω ) v (0) = F ( v (0)) ⇔ m − ω ) = F ( v (0)) v (0) . We define the function G ∈ C ([0 , + ∞ )) ∩ C ((0 , + ∞ )) by G ( s ) = F ( s ) /s on R + . Asconsequences of Assumption 1.1, we have that G (0) = 0, since0 < F ( s ) = Z s f ( t ) d t < f ( s ) Z s d t = sf ( s ) , ∀ s > . Also, lim s → + ∞ G ( s ) = + ∞ , since lim s → + ∞ f ( s ) = + ∞ and G ( s ) > s Z ss/ f ( t ) d t > f ( s/ / . Finally, G is strictly increasing on R + since s G ′ ( s ) = sf ( s ) − F ( s ) > , ∀ s > . Thus, G has a continuous inverse on R + and v (0) = G − (cid:0) m − ω ) (cid:1) . Since G − isincreasing and positive, it has a limit at zero and, since G (0) = 0, this limit necessarilyequals zero. Thus lim ω → m v (0) = 0, as expected.Also note that this equation implies that u ( x ) = 0 if x = 0, since otherwise therewould be x > v ( x ) , u ( x )) = ( v (0) , u (0)) and the solution would be periodicin x , contradicting φ ∈ L ( R ). On one hand, this implies that v > u > R \ { } hence that v > R because v (0) > v is continuous. On another hand, weobserve that the eigenvalue equation (3) is equivalent to ( v ′ = − ( M + ω ) uu ′ = − ( M − ω ) v . Therefore, using definition (12) of M and since we obtained that f > G on R + , we have u ′ (0) = ( ω − m + f ( v (0))) v (0) > (cid:0) ω − m + G (cid:0) v (0) (cid:1)(cid:1) v (0) = ( m − ω ) v (0) > . Finally, since u is continuous, non-zero on R + with u ′ (0) >
0, we conclude that u > R + . N SPECTRAL STABILITY OF SOLER STANDING WAVES 13
The eigenvalue equation also implies( v − u ) ′ = 2 uv ( − M − ω + M − ω ) = − ωuv . Therefore, since v, u > R + , v − u is decreasing on R + and has its maximum at 0.We now use (11) to write || Q || ( x ) mω (cid:0) v − u (cid:1) ( x ) f ′ (cid:0)(cid:0) v − u (cid:1) ( x ) (cid:1) mω sup s ∈ [0 ,v (0)] sf ′ ( s ) . By Assumption 1.1, we concludelim ω → m ||| Q ||| mω lim ω → m sup s ∈ [0 ,v (0)] sf ′ ( s ) = 0 . (cid:3) Bifurcations from the origin
The goal of this section is to prove Theorem 1.4. We first recall some basic facts aboutperturbation theory for closed operators. Proofs can be found in standard referencessuch as [19, 21]. The algebraic multiplicity m a ( λ ) of an isolated point λ of the spectrumof a closed operator T is defined as the dimension of the range of the Riesz projector P γ ( T ) = 12 πi Z γ ( T − z ) − d z, where γ is any closed, simple contour such that γ ⊂ ρ ( T ) and λ is the only point ofthe spectrum of T inside γ . This multiplicity coincides, see [19, Chapter 6], with thedimension of the generalized eigenspace ∪ n ∈ N ker( T − λ ) n . The geometric multiplicity m g ( λ ) is the dimension of ker( T − λ ). From this, it followsthat m a ( λ ) > m g ( λ ), and if m a ( λ ) >
0, then m g ( λ ) >
0. If T is self-adjoint, the geomet-ric and algebraic multiplicities of any eigenvalue coincide. The algebraic multiplicity isthe correct multiplicity for analytic perturbation theory for closed operators.Recall that a family of operators µ T µ is said to be a type-A analytic family in anopen set S ⊂ C if for all µ ∈ S the operator T µ is closed on a domain D independentof µ , and if for each ϕ ∈ D the map µ T µ ϕ is strongly analytic. For shortness, wecall such family { T µ } µ an analytic family .If { T µ } µ is an analytic family of operators and γ is a contour such that γ ⊂ ρ ( T µ ) forany µ ∈ B (0 , r ), then the sum of algebraic multiplicities of eigenvalues in the interiorof γ is constant for all µ ∈ B (0 , r ), see [19, Chapter 15].In our particular case, the families { L µ } µ ∈ C and { H µ } µ ∈ C are linear in µ and thereforeanalytic. They have isolated eigenvalues of finite algebraic multiplicity away from theiressential spectra. By analytic perturbation theory, the eigenvalue branches λ ( µ ) of L µ are analytic functions of µ , and those of H µ are continuous. We start this section with a lemma that characterizes Condition i) from Theorem 1.4in terms of the invertibility of L µ . Lemma 3.1.
Let f satisfy Assumption 1.1. Then the following statements are equiva-lent Analytic functions of µ as long as eigenvalues are simple, and branches of a multivalued functionanalytic in ( µ − µ ) /k if at least k ∈ N branches intersect for µ . i) L has a single eigenvalue in ( − ω, ,ii) / ∈ σ ( L µ ) for all µ ∈ (0 , .Proof. Recall that eigenvalues of L µ are analytic in µ . Let λ ( µ ) be an eigenvalue of L µ and ϕ ( µ ) be a corresponding normalized eigenfunction, then first order perturbationtheory (sometimes known as the Feynman–Hellmann theorem), together with the non-negativity of Q , yields ∂ µ λ ( µ ) = − h ϕ ( µ ) , Qϕ ( µ ) i . (14)Denote by λ ( µ ) the branch with λ (0) = 0 and observe that ∂ µ λ (0) = − (cid:10) v − u , f ′ ( v − u ) (cid:0) v − u (cid:1)(cid:11) < , by Assumption 1.1 and the definition of Q . Thus, λ ( µ ) < µ >
0. Moreover, λ ( µ ) > − ω since − ω is an eigenvalues of L µ for any µ > ∂ µ λ ( µ ) − ω < λ ( µ ) < , for µ > . Therefore, the number of eigenvalues of L in ( − ω,
0) equals the number of µ ’s in [0 , L µ . (cid:3) Now we can prove our first main result.
Proof of Theorem 1.4.
We denote by m a ( λ, H µ ) the algebraic multiplicity of λ as aneigenvalue of H µ . For µ = 0, the operator H is self-adjoint and we have m a (0 , H ) = 2since ker( H n ) = ker( H ) = span (cid:26)(cid:18) φ (cid:19) , (cid:18) φ (cid:19)(cid:27) . In view of Lemma 3.1, the function l ( µ ) := (cid:10) φ , L − µ φ (cid:11) is well defined on (0 , µ ∈ (0 , m a (0 , H µ ) > l ( µ ) = 0. Indeed,ker( H µ ) = ker( L L µ ) × ker( L µ L ) = span (cid:26)(cid:18) L − µ φ (cid:19) , (cid:18) φ (cid:19)(cid:27) , so m a (0 , H µ ) >
2. Now fix µ ∈ (0 , ψ , ψ ) T ∈ ker( H µ ). Thecorresponding equations are (cid:26) L µ L L µ ψ = 0 ,L L µ L ψ = 0 . Since L µ is invertible, the first equation implies ψ ∈ ker( L L µ ). Since ker( L ) =span( φ ), the second equation implies that L µ L ψ = αφ for some α ∈ C . If α = 0, we conclude that ψ ∈ ker( L µ L ) andker( H µ ) ⊂ ker( H µ ) . By induction on n , this implies thatker( H nµ ) ⊂ ker( H µ ) N SPECTRAL STABILITY OF SOLER STANDING WAVES 15 for all n ∈ N , and m a (0 , H µ ) = dim(ker( H µ )) = 2. Thus, if m a (0 , H µ ) >
3, we need α = 0 and, taking the inner product of both sides of L ψ = αL − µ φ with φ yields 0 = α (cid:10) φ , L − µ φ (cid:11) = α l ( µ ) , hence l ( µ ) = 0 as claimed. For the converse, if l ( µ ) = 0, then L − µ φ ∈ ker( L ) ⊥ = ran( L )and thus there exists ˜ φ / ∈ ker( L ) such that L L µ L ˜ φ = 0 . Since again, ker( H µ ) = ker( L µ L L µ ) × ker( L L µ L ) , this implies m a (0 , H µ ) > ∂ ω || φ ( ω ) || L , (15)then l ( µ ) < µ ∈ (0 , l is non-decreasing, since l ′ ( µ ) = lim h ց h − (cid:10) φ , ( L − µ + h − L − µ ) φ (cid:11) = lim h ց (cid:10) φ , L − µ + h QL − µ φ (cid:11) = (cid:10) L − µ φ , QL − µ φ (cid:11) > . Next, we note that l is continuous as µ →
2, since ker( L ) = ∂ x φ and φ is orthogonalto ∂ x φ . We conclude that φ ∈ ran( L ) and there exists a unique ψ such that L ψ = φ and h ψ , ∂ x φ i = 0 . We obtain l (2) := lim µ ր l ( µ ) = h φ , ψ i . Finally, we use the identity L ∂ ω φ = φ , which can be obtained by taking the derivative with respect to ω of the equation L φ =0, to conclude ψ = ∂ ω φ and l (2) = h φ , ∂ ω φ i = 12 ∂ ω || φ || . If (15) holds with a strict inequality, this is sufficient to conclude l ( µ ) l (2) < µ ∈ (0 , m a (0 , H µ ) = 2.If the equality holds in (15), we have to show that l is not constant on any interval[ µ , l is real analytic, this can only happen if l ( µ ) = 0 for all µ ∈ (0 , m a (0 , H µ ) > µ ∈ (0 , µ , since m a (0 , H ) = 2 and || H − H µ || = µ ||| Q ||| . Let δ = dist(0 , σ ( H ) \ { } ) >
0. Then thecircle ∂B (0 , δ/
2) is contained in the resolvent set ρ ( H µ ) for all µ with | µ | < δ/ (2 ||| Q ||| ).Indeed, for all φ ∈ H ( R , C ) and z ∈ ∂B (0 , δ/ || ( H µ − z ) φ || > || ( H − z ) φ || − || ( H − H µ ) φ || > (cid:18) δ − µ ||| Q ||| (cid:19) || φ || . Thus, by [19], the sums of algebraic multiplicities of the eigenvalues of H µ within B (0 , δ/
2) is constant for µ ∈ [0 , δ/ (2 ||| Q ||| )) and therefore m a (0 , H µ ) = 2 for thosevalues of µ . (cid:3) Corollary 3.2.
Assume that the conditions of Theorem 1.4 hold and that for any z ∈ C \ i R eigenvalue of H µ the following holds inf µ ∈ [0 , Re z > . Then, H has no non-zero eigenvalues on the imaginary axis.Proof. Recall that H has only real eigenvalues. Consider the eigenvalues of H µ as µ increases from 0 to 2. Since H µ is an analytic family andRe z > ⇔ | Im z | | Re z | , any branch of eigenvalues that goes to the imaginary axis must pass through zero. Butthis is not possible since, by Theorem 1.4, the algebraic multiplicity of zero, as aneigenvalue of H µ , is constant (equal to 2) for µ ∈ (0 , (cid:3) We end this section with a by-product of the proof of Lemma 3.1, that we will needfor the proof of Theorem 7.2, about the smallest eigenvalue of L µ (strictly) above − ω . Lemma 3.3.
Let f satisfy Assumption 1.1 and µ > . Then the first eigenvalue λ of L µ in ( − ω, m − ω ) exists and verifies − µ ||| Q ||| λ ( µ ) < . Moreover, if − ω and are the only eigenvalues of L , then for any µ > , L µ hasno eigenvalues in the interval ( − m − ω, − ω ) .Proof. The argument in the proof of Lemma 3.1 gives the existence and the negativity.Moreover, integrating (14), for λ , over µ gives λ ( µ ) = 0 − Z µ h φ ν , Qφ ν i d ν > − µ ||| Q ||| , where φ ν is the normalized eigenvector associated to λ ( ν ).The second result is obtained by the same Feynman–Hellmann type argument as inthe proof of Lemma 3.1. (cid:3) Lower bound on Re z In this section, our goal is to establish bounds for eigenvalues of H µ away from thereal and imaginary axes. These bounds say that, for sufficiently small ||| Q ||| , theseeigenvalues can only occur close to the outer thresholds ± ( m + ω ). Throughout thissection we assume that z ∈ C is an eigenvalue with associated eigenvector ϕ = ( ϕ , ϕ ) T ,and ϕ , ϕ are two-component spinors. We introduce the notations P − = E ( −∞ , − ω ) ( L ) and P + = E (0 , ∞ ) ( L )for the spectral projectors of L on the corresponding intervals. Note that the eigen-functions associated to − ω and 0 are excluded from the range of these projectors. Fora concise notation in the proofs below, we also define Q ++ = P + QP + , Q −− = P − QP − , and Q + − = Q ∗− + = P + QP − . N SPECTRAL STABILITY OF SOLER STANDING WAVES 17
We first establish an identity for eigenvalues.
Lemma 4.1.
Let f satisfy Assumption 1.1. Assume that z ∈ C \ ( R ∪ i R ) is aneigenfunction of H µ with eigenfunction ( ϕ , ϕ ) T . Then ϕ = ( P + + P − ) ϕ , h ϕ , ϕ i = 0 and Re z = (cid:10) ϕ , (cid:0) P + L µ P + − P − L µ P − (cid:1) ϕ (cid:11)(cid:10) ϕ , | L | − ϕ (cid:11) , (16) where L − ϕ is well-defined since ϕ ∈ ker( L ) ⊥ .Proof. Since there are no eigenvalues of L in ( − ω,
0) by Theorem 1.5, we only have tocheck that ϕ is orthogonal to the eigenfunctions ϕ and ϕ − ω (associated to 0 and − ω )in order to prove ϕ = ( P + + P − ) ϕ . Taking the inner product of (13a) with ϕ gives z h ϕ , ϕ i = 0. Since z = 0, this gives the first ortogonality condition. The eigenvalueequation for H µ reads ( L µ L ϕ = z ϕ L L µ ϕ = z ϕ . (17)Taking the inner product of the first line with ϕ − ω , and using that L µ ϕ − ω = L ϕ − ω gives z h ϕ − ω , ϕ i = 4 ω h ϕ − ω , ϕ i . Since we assume z to not be real, this implies the orthogonality of ϕ to ϕ − ω .To prove h ϕ , ϕ i = 0, we start by taking the scalar product of (13a) and (13b)respectively with ϕ and ϕ . We obtain (cid:26) h ϕ , L ϕ i = z h ϕ , ϕ ih ϕ , L µ ϕ i = z h ϕ , ϕ i . By selfadjointness of L and L µ , the left hand side of both equations is real. Takingtheir sum and difference gives (cid:26) z Re h ϕ , ϕ i = h ϕ , L ϕ i + h ϕ , L µ ϕ i ∈ R iz Im h ϕ , ϕ i = h ϕ , L ϕ i − h ϕ , L µ ϕ i ∈ R . The first line yields Re h ϕ , ϕ i = 0, since z R , and the second gives Im h ϕ , ϕ i = 0,since z i R .We now turn to the proof of (16). We apply L − to the second line of (17) and takethe scalar product with P + ϕ and P − ϕ . This gives ( h P + ϕ , L µ ϕ i = z (cid:10) P + ϕ , L − ϕ (cid:11) h P − ϕ , L µ ϕ i = z (cid:10) P − ϕ , L − ϕ (cid:11) . We insert ϕ = ( P + + P − ) ϕ , and use the identity P + L µ P − = P + L P − − µP + QP − = − µQ + − . We are left with ( h ϕ , P + L µ P + ϕ i − µ h ϕ , Q + − ϕ i = z (cid:10) ϕ , P + L − P + ϕ (cid:11) h ϕ , P − L µ P − ϕ i − µ h ϕ , Q − + ϕ i = z (cid:10) ϕ , P − L − P − ϕ (cid:11) . Subtracting both identities yields h ϕ , ( P + L µ P + − P − L µ P − ) ϕ i − µ h ϕ , ( Q + − − Q − + ) ϕ i = z (cid:10) ϕ , | L | − ϕ (cid:11) . Taking the real part eliminates the second term in the l.h.s. and gives identity (16). (cid:3)
We now exploit this identity to derive a lower bound on Re z . This bound dependson t := min { λ | λ ∈ σ ( L + ω ) ∩ ( ω, m ] } , (18)which is either the smallest eigenvalue of L + ω above ω , or the bottom of the positiveessential spectrum at m . Lemma 4.2.
Let f satisfy Assumption 1.1, µ > , α ∈ [0 , , and t be defined asin (18) . Assume that z ∈ C \ ( R ∪ i R ) is an eigenfunction of H µ with eigenfunction ( ϕ , ϕ ) T , and define η ∈ R + as η := || P + ϕ || || P − ϕ || . If (cid:0) − α (cid:1) η + (1 + α ) > µη ||| Q ||| t + ω , (19) then Re z > ( t + ω ) (cid:18) (1 − α ) η + (1 + α ) − η α µ ||| Q ||| t + ω (cid:19) > . (20) Proof.
By combining the eigenvalue equations (13a) and (13b) with h ϕ , ϕ i = 0, weobtain h ϕ , L µ ϕ i = 0 , and (cid:10) ϕ , L − ϕ (cid:11) = 0 . (21)For shortness, we define ϕ + := P + ϕ and ϕ − := P − ϕ . The above identities give h ϕ + , L µ ϕ + i + h ϕ − , L µ ϕ − i − µ h ϕ , ( Q + − + Q − + ) ϕ i = 0 (22)and (cid:10) ϕ + , | L | − ϕ + (cid:11) − (cid:10) ϕ − , | L | − ϕ − (cid:11) = 0 . (23)As a first consequence of these identities, ϕ + = 0 and ϕ − = 0, hence η ∈ R + is well-defined.From the symmetry w.r.t. 0 of the spectrum of L + ω and the definition of t , wecompute inf ψ ∈ Ran( P − ) h ψ, | L | ψ i|| ψ || = inf ψ ∈ Ran( P − ) h ψ, − ( L + ω ) ψ i|| ψ || + ω = inf ψ ∈ Ran( P + ) h ψ, ( L + ω ) ψ i|| ψ || + ω = t + ω, and similarly for | L | − . Therefore, h ϕ − , | L | ϕ − i > ( t + ω ) || ϕ − || and (cid:10) ϕ − , | L | − ϕ − (cid:11) ( t + ω ) − || ϕ − || . (24)We now bound all quantities appearing in (16). For the denominator, combining (23)with (24) gives (cid:10) ϕ , | L | − ϕ (cid:11) = 2 (cid:10) ϕ − , | L | − ϕ − (cid:11) t + ω ) − || ϕ − || . N SPECTRAL STABILITY OF SOLER STANDING WAVES 19
The first term in the numerator has to be treated in two different ways depending onthe value of η . From (22), we obtain h ϕ + , L µ ϕ + i = − h ϕ − , L µ ϕ − i + µ h ϕ , ( Q + − + Q − + ) ϕ i , which will be a useful identity to obtain bounds for small values of η .On the other hand, for large values of η , we apply Jenssen’s inequality (to the spectralmeasure of L ) to obtain h ϕ + , L ϕ + i|| ϕ + || > (cid:10) ϕ + , L − ϕ + (cid:11) || ϕ + || ! − = || ϕ + || (cid:10) ϕ − , | L | − ϕ − (cid:11) , where we have also used (23). Combining with (24) and the definition of η , we obtain h ϕ + , L µ ϕ + i = h ϕ + , L ϕ + i − µ h ϕ + , Qϕ + i > || ϕ + || ( t + ω ) || ϕ − || − − µ h ϕ + , Qϕ + i = η ( t + ω ) || ϕ − || − µ h ϕ + , Qϕ + i . Inserting these two statements on h ϕ + , L µ ϕ + i into (16), we obtain2 (cid:10) ϕ − , | L | − ϕ − (cid:11) Re z = α h ϕ + , L µ ϕ + i + (1 − α ) h ϕ + , L µ ϕ + i − h ϕ − , L µ ϕ − i > α ( − h ϕ − , L µ ϕ − i + µ h ϕ , ( Q + − + Q − + ) ϕ i )+ (1 − α ) (cid:0) η ( t + ω ) || ϕ − || − µ h ϕ + , Qϕ + i (cid:1) − h ϕ − , L µ ϕ − i , for all α ∈ [0 , (cid:10) ϕ − , | L | − ϕ − (cid:11) Re z > (cid:0) α (1 − η ) + 1 + η (cid:1) ( t + ω ) || ϕ − || + µ h ϕ , ((1 + α ) Q −− + αQ + − + αQ − + − (1 − α ) Q ++ ) ϕ i . In the last term, we “complete the square” to obtain (cid:10) ϕ, (cid:0) (1 + α ) Q −− + αQ + − + αQ − + − (1 − α ) Q ++ (cid:1) ϕ (cid:11) = (1 + α ) (cid:10) ϕ − + α α ϕ + , Q (cid:0) ϕ − + α α ϕ + (cid:1)(cid:11) − α α h ϕ + , Qϕ + i − (1 − α ) h ϕ + , Qϕ + i > − (cid:16) α α + (1 − α ) (cid:17) ||| Q ||| || ϕ + || = − η α ||| Q ||| || ϕ − || . This finally gives2 (cid:10) ϕ − , | L | − ϕ − (cid:11) || ϕ − || Re z > (cid:0) α (1 − η ) + 1 + η (cid:1) ( t + ω ) − µη α ||| Q ||| , ∀ α ∈ [0 , . Now, if (19) —in Lemma 4.2— holds, i.e., if the r.h.s. is nonnegative, then the l.h.s.verifies 2( t + ω ) − Re z > (cid:10) ϕ − , | L | − ϕ − (cid:11) || ϕ − || Re z > z > ( t + ω ) (cid:18) (1 − α ) η + (1 + α ) − η α µ ||| Q ||| t + ω (cid:19) > , ∀ α ∈ [0 , . This concludes the proof of the lemma. (cid:3)
The following technical lemma is the last step to the proof of Theorem 1.6 and willalso be used to obtain explicit bounds in the next section.
Lemma 4.3.
Let f satisfy Assumption 1.1, t be defined as in (18) , and θ + : [0 , → (0 , √ / as in (27) below. Assume that ω ∈ (0 , m ) , E ∈ [0 , m ) , and µ > are suchthat E < t + ω and µ ||| Q ||| t + ω ) < θ + (cid:18) E ( t + ω ) (cid:19) . If z ∈ C \ ( R ∪ i R ) is an eigenvalue of H µ , then Re z > E . Proof.
We start from (20), where we introduce the parameter θ := µ ||| Q ||| t + ω ) for shortness,and study the function g θ ( α, η ) := (1 − α ) η + (1 + α ) − ηθ α . Since we do not know the value of η >
0, we need to study the question: Given ξ > θ > η> max α ∈ [0 , g θ ( α, η ) > ξ is verified?Notice that ∂ α g θ ( α, η ) = − η + 4 θ (1 + α ) η + 1is a decreasing function of α > η >
0. Thus, ∂ α g θ ( α, η ) > ∂ α g θ (1 , η ) = − η + θη + 1 . Defining η ⋆ as the positive number such that − η ⋆ + θη ⋆ + 1 = 0, i.e., η ⋆ = θ + √ θ + 42 > , we have for all η ∈ (0 , η ⋆ ] that ∂ α g θ ( α, η ) > − η + θη + 1 > − η ⋆ + θη ⋆ + 1 = 0 . Thus, for η ∈ (0 , η ⋆ ], α g θ ( α, η ) is an increasing function on [0 ,
1] and its maximumis g θ (1 , η ) = 2(1 − ηθ ).Similarly, for any α > ∂ α g θ ( α, η ) ∂ α g θ (0 , η ) = − η + 4 θη + 1 . Defining η ◦ as the positive number such that − η ◦ +4 θη ◦ +1 = 0, i.e., η ◦ := 2 θ + √ θ + 1,we have for all η > η ◦ that ∂ α g θ ( α, η ) − η + 4 θη + 1 − η ◦ + 4 θη ◦ + 1 = 0 . Thus, for η > η ◦ , α g θ ( α, η ) is a decreasing function on [0 ,
1] and its maximum is g θ (0 , η ) = η − θη + 1. N SPECTRAL STABILITY OF SOLER STANDING WAVES 21
Finally, for η ∈ ( η ⋆ , η ◦ ), α g θ ( α, η ) is increasing then decreasing on [0 , α + := q ηθη − − h ( η ) := g θ ( α + , η ) = (cid:16) η − √ θ p η ( η − (cid:17) . Now, since η g θ (0 , η ) = η − θη + 1 > η ◦ , + ∞ ), η g θ (1 , η ) = 2(1 − ηθ )is decreasing ( θ > h ( η ⋆ ) = 2(1 − η ⋆ θ ) by construction, and (using the identity η ⋆ = θη ⋆ + 1) h ′ ( η ⋆ ) = 2 η ⋆ − √ θ η ⋆ − p η ⋆ ( η ⋆ − ! = − θ < , we conclude thatinf η> max α ∈ [0 , g θ ( α, η ) = inf η ⋆ <η<η ◦ h ( η ) = h ( η θ ) = 2 η θ − η θ η θ − , where η θ is defined as the unique real number in (1 , + ∞ ) such that4 η θ ( η θ − η θ − = θ. (25)Note that the l.h.s. of (25) being a strictly increasing function on (1 , + ∞ ) —henceone-to-one from (1 , + ∞ ) to (0 , + ∞ ) ∋ θ —, it gives that (25) has a unique solution η θ in (1 , + ∞ ) and, on another hand that η ⋆ < η θ < η ◦ . Indeed, recalling that 1 < η ⋆ < η ◦ and the equations they respectively solve, we have η ⋆ < η θ ⇔ θ > η ⋆ ( η ⋆ − η ⋆ − = 4 η ⋆ (3 η ⋆ − θ ⇔ η ⋆ > η θ < η ◦ ⇔ θ < η ◦ ( η ◦ − η ◦ − = 16 η ◦ (3 η ◦ − θ ⇔ η ◦ − < η ◦ . Moreover, we obtain, as a by-product, that the problem has a solution only for ξ < η> max α ∈ [0 , g θ ( α, η ) = inf η ⋆ <η<η ◦ h ( η ) = h ( η ⋆ ) = 2(1 − η ⋆ θ ) < . Summarizing, we have obtained η θ > η> max α ∈ [0 , g θ ( α, η ) > ξ ⇔ h ( η θ ) > ξ. Now, keeping in mind that ξ ∈ [0 , η η − η η − is strictly decreasing andthat η η θ ( η θ − η θ − is stritly increasing, we haveinf η> max α ∈ [0 , g θ ( α, η ) > ξ ⇔ h ( η θ ) > ξ ⇔ η θ q − ξ + p − ξ ) + 8 ξ ⇔ µ ||| Q ||| t + ω ) =: θ θ + ( ξ ) , (26) where θ + : [0 , → (0 , √ /
8] is defined by θ + ( ξ ) := 2 (cid:16) − ξ + p − ξ ) + 8 ξ (cid:17) (cid:16) − ξ + p − ξ ) + 8 ξ (cid:17) (cid:16) − ξ + 3 p − ξ ) + 8 ξ (cid:17) . (27) (cid:3) Remark 4.4.
The fact that the problem has a solution only for ξ < , ( t + ω ) ). That is, we cannot reach the innerthreshold ± ( m + ω ), even when t = m . Remark 4.5.
Expression (27) might seem complicated but in concrete cases, one simplyevaluates it at some ξ ∈ [0 ,
2) and obtains just a number.We now complete the proof of Theorem 1.6.
Proof of Theorem 1.6.
In view of Lemma 2.4, we only have to prove (8). First, if E = 0,since lim ω → m ||| Q ω ||| = 0 from Lemma 2.4, there exists ω E =0 ∈ (0 , m ) such that for all µ ∈ (0 ,
2] and ω ∈ [ ω E =0 , m ), µ ||| Q ω ||| t + ω ) θ + (0) = 3 √ . By Lemma 4.3, this implies (8) in the case E = 0.Now, for a fixed E ∈ (0 , m ), we restrict our attention to ω ’s such that ω ∈ [ E/ , m ).In this case, we choose ξ ≡ ξ ( E, ω, t ) as E m ξ := 2 E ( t + ω ) < E ω , so we obtain lim ω → m ξ ( E, ω, t ) = E / (2 m ) <
2. Since lim ω → m ||| Q ω ||| = 0 fromLemma 2.4, there exists ω E ∈ [ E/ , m ) such that for all µ ∈ (0 ,
2] and ω ∈ [ ω E , m ), µ ||| Q ω ||| t + ω ) ||| Q ω ||| t + ω ) θ + ( ξ ) . By Lemma 4.3, this implies Re z > ( t + ω ) ξ = E . This concludes the proof of Theorem 1.6. (cid:3)
These results show that, in order to obtain quantitative bounds, we need to estimateor compute ||| Q ||| . This is done in the next section for pure power nonlinearities. Weonly know the value of t from (18) for the case p = 1, but for the general case we cansimply bound t > ω . N SPECTRAL STABILITY OF SOLER STANDING WAVES 23 Power nonlinearities: explicit estimates
From now on, we consider the case of a power nonlinearity: f ( s ) = | s | p , p >
0, whichverifies Assumption 1.1. In that case, the solitary wave solutions φ ( x ) := φ ( p, ω ; x ) := (cid:18) v ( p, ω ; x ) u ( p, ω ; x ) (cid:19) , which verify v − u >
0, as recalled in (4), can be found by explicitly integrating theODE, see e.g. [9, 11, 24, 25], and are given for any ( p, ω ) ∈ (0 , + ∞ ) × (0 , m ) by v ( x ) := v ( p, ω ; x ) := 1 q − ν tanh ( pκx ) (cid:20) ( p + 1)( m − ω ) 1 − tanh ( pκx )1 − ν tanh ( pκx ) (cid:21) p (28)and u ( x ) := u ( p, ω ; x ) := √ ν tanh( pκx ) v ( p, ω ; x ) , (29)where we have introduced the parameters κ = √ m − ω and ν = m − ωm + ω ∈ (0; 1) . These explicit formulae allow us to compute exactly the contribution of φ in L : f (cid:0) v − u (cid:1) = (cid:0) v − u (cid:1) p = ( p + 1)( m − ω ) 1 − tanh ( pκ · )1 − ν tanh ( pκ · ) (30)= 2 m ( p + 1) ν ν − tanh ( pκ · )1 − ν tanh ( pκ · ) , which we wrote in two ways as the latter point of view will turn out to be useful insome of our proofs. We already and immediately read from it that (cid:12)(cid:12)(cid:12)(cid:12) f (cid:0) v − u (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ ( R ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) v − u (cid:1) p (cid:12)(cid:12)(cid:12)(cid:12) L ∞ ( R ) = ( p + 1)( m − ω ) = 2 m ( p + 1) ν ν . (31)Inserting (28) and (29) in the definition of Q in (6) leads to the explicit expression Q = p (cid:0) v − u (cid:1) p − (cid:18) v − uv − uv u (cid:19) = p ( p + 1)( m − ω ) 1 − tanh ( pκ · ) (cid:0) − ν tanh ( pκ · ) (cid:1) (cid:18) −√ ν tanh( pκ · ) −√ ν tanh( pκ · ) ν tanh ( pκ · ) (cid:19) . (32)5.1. Rescaled operators.
Given the explicit spatial depends in pκx of φ and becauseit will often be convenient for shortness and clarity, we will use the convention that atilde means a spatial rescaling by a factor pκ . That is, for instance˜ v ( x ) = ˜ v ( p, ω ; x ) = 1 q − ν tanh ( x ) (cid:20) ( p + 1)( m − ω ) 1 − tanh ( x )1 − ν tanh ( x ) (cid:21) p or, more concisely,˜ v = 1 p − ν tanh (cid:20) ( p + 1)( m − ω ) 1 − tanh − ν tanh (cid:21) p . Analoguously, we also define the operator U pκ through its adjoint U ∗ pκ ψ ( x ) := √ pκψ ( pκx ) , and, for any operator O , the corresponding operator ˜ O := U pκ OU ∗ pκ . For instance,˜ D m := U pκ D m U ∗ pκ = ipκσ ∂ x + mσ ,˜ L µ := U pκ L µ U ∗ pκ := ˜ L µ ( p, ω ) := ipκσ ∂ x + mσ − ω −|h e φ , σ e φ i| p σ − µ ˜ Q with ˜ Q := U pκ QU ∗ pκ = p ( p + 1)( m − ω ) 1 − tanh (cid:0) − ν tanh (cid:1) (cid:18) −√ ν tanh −√ ν tanh ν tanh (cid:19) , and ˜ H µ := U pκ H µ U ∗ pκ = (cid:18) L ˜ L µ (cid:19) . This unitary operator U pκ leaves invariant spectra and transforms eigenfunctions intoeigenfunctions.With these explicit formulae and notations in place, we can start computations. Wefirst check the Vakhitov–Kolokolov criterion, then we compute ||| Q ( p, ω, · ) ||| and plug itinto the bounds in Section 4, to prove Theorem 1.7.5.2. Vakhitov–Kolokov condition.
The following lemma gives, in the case of powernonlinearities, sufficient conditions on ( p, ω ) for the Vakhitov–Kolokov condition (15)to hold as well as sufficient conditions for it to not hold.
Lemma 5.1.
Let f ( s ) = | s | p . • If p ∈ (0 , , then ∀ ω ∈ (0 , m ) , ∂ ω || φ ( p, ω ; · ) || L < . • If p > , then ∀ ω p p − m, ∂ ω || φ ( p, ω ; · ) || L < . and there exists ω + ∈ (cid:16) p p − m, q p +12 p − m i such that ∀ ω > ω + , ∂ ω || φ ( p, ω ; · ) || L > . Proof of Lemma 5.1.
First, since ∂ ω ν = − m ( m + ω ) <
0, we have thatsign ∂ ω || φ ( p, ω ; · ) || L = − sign ∂ ν || φ ( p, ω ; · ) || L and we work with the latter in this proof.Using (28), (29), ( p + 1)( m − ω ) = 2 m ( p + 1) ν ν and κ = m ν √ ν , we have || φ || L = (cid:12)(cid:12)(cid:12)(cid:12) v + u (cid:12)(cid:12)(cid:12)(cid:12) L = 1 pκ (cid:12)(cid:12)(cid:12)(cid:12) ˜ v + ˜ u (cid:12)(cid:12)(cid:12)(cid:12) L = 2 pκ Z ∞ ˜ v + ˜ u = m p − p ( p + 1) p p ν √ ν (cid:18) ν ν (cid:19) p Z ∞ ν tanh − ν tanh (cid:20) − tanh − ν tanh (cid:21) p N SPECTRAL STABILITY OF SOLER STANDING WAVES 25 =: m p − p ( p + 1) p p F ( ν ) , and our goal is to determine the sign of ∂ ν F ( ν ). We have F ( ν ) = 1 + ν √ ν (cid:18) ν ν (cid:19) p Z νy − νy (cid:20) − y − νy (cid:21) p d y − y =: Z h ( ν, y ) d y . Since y (1 + νy )(1 − νy ) − (1+ p − ) is increasing on (0 ,
1) for any ν ∈ (0 , < h ( ν, y ) ν p − (1 + ν ) − p (1 − ν ) p (1 − y ) p − ∈ L ((0 , , because R (1 − y ) p − d y = √ π p )Γ( + p ) < + ∞ for any p >
0. Hence y h ( ν, y ) isLebesgue intgrable for any ν ∈ (0 , y ∈ (0 , ∂ ν h ( ν, y ) exists forall ν ∈ (0 ,
1) and ∂ ν h ( ν, y ) = ν p − (1 + ν ) p × P ( y )2(1 − νy ) p +2 (1 − y ) − p , where the polynomial P ν ( z ) := ν (cid:18) p ν + (1 − ν ) (cid:19) z + ν (1 + ν ) (cid:18) p + 4 (cid:19) z + (cid:18) ν + 2 p − (cid:19) is strictly increasing on (0 , | ∂ ν h ( ν, y ) | ν p − | P ν (1) | − ν ) p +2 (1 + ν ) p × − y ) − p , which is in L ((0 , y , for any ν ∈ (0 , F ′ ( ν ) = ∂ ν Z h ( ν, y ) d y = Z ∂ ν h ( ν, y ) d y . Now, a sufficient condition for ∂ ω || φ || L < ∂ ν || φ || L > P ν to be positive on (0 ,
1) or equivalently, since P ν is strictly increasing on (0 , P (0) = ν + p − >
0. This proves the claim in the case p p > ν > − p ⇔ ω mp − .Similarly, a sufficient condition for ∂ ω || φ || L > P ν to benegative on (0 ,
1) or, equivalently, that P (1)
0. Assuming p >
2, and since 0 < ν < P (1) ⇔ (1 + ν ) (cid:0) ( p − ν − pν + p − (cid:1) > ⇔ ν ν + ( p ) , with ν + ( p ) := 3 pp − − p − p (2 p − p + 1) . Defining ω + := − ν + ν + m = q p +12 p − m , we have proved the second claim in the case p > Now, for the first claim in the case p > ω ∈ (cid:16) mp − , p p − m i ⇔ ν ∈ h p − p − , − p (cid:17) ,we proceed as follow. Since the function y ∂ ν f ( ν, y ) is strictly increasing on (0 , ν , we have F ′ ( ν ) = Z z ∂ ν h ( ν, y ) d y + Z z ∂ ν h ( ν, y ) d y > z∂ ν h ( ν,
0) + (1 − z ) ∂ ν h ( ν, z ) . Inserting z = √ ν in the r.h.s., we obtainr.h.s. = ν p − (1 + ν ) p √ ν (cid:18) ν + 2 p − (cid:19) + (cid:16) p ν + 1 − ν (cid:17) ν + ν (1 + ν ) (cid:16) p + 4 (cid:17) + (cid:16) ν + p − (cid:17) (1 + √ ν )(1 + ν ) p (1 − ν ) and a sufficient condition to ensure F ′ ( ν ) > ∂ ω || φ || L <
0— is (cid:18) p − (cid:19) ν + (cid:18) p + 5 (cid:19) ν + (cid:18) p + 4 (cid:19) ν + n √ ν (1 + √ ν )(1 + ν ) p (cid:0) − ν (cid:1) o (cid:18) ν + 2 p − (cid:19) > . Now, since we are in the case ν + p − <
0, noticing that0 < √ ν (1 + √ ν )(1 + ν ) p (cid:0) − ν (cid:1) < p < √ < , ∀ ν ∈ (0 , (cid:18) p − (cid:19) ν + (cid:18) p + 5 (cid:19) ν + (cid:18) p + 4 (cid:19) ν + 3 ν + 3 (cid:18) p − (cid:19) > . Finally, this polynomial in ν being strictly increasing on (0 ,
1) and positive at ν = p − p − concludes the proof that ∂ ω || φ || L < p > ω ∈ (cid:16) mp − , p p − m i . (cid:3) Estimates on β ( p ) . We derive here lower bounds on ω for condition (26) to hold.We do that by means of our general bounds on Re z combined with the explicit formulaof the norm ||| Q ω ||| of Q ω . Lemma 5.2.
The operator Q ω , which acts as point-wise multiplication by the two bytwo matrix given in (32) , is positive semi-definite and satisfies ||| Q ω ||| = p ( p + 1) m ν ν = p ( p + 1)( m − ω ) , if ν < ⇔ ω > m ,p p + 12 m ν − ν = p p + 12 m ω , if ν > ⇔ ω m . (33) Proof.
For all x ∈ R , the matrix Q ω ( x ) is positive semi-definite with eigenvalues 0and p ( v − u ) p − ( v + u ), hence ||| Q ω ||| = p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) v − u (cid:1) p − (cid:0) v + u (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ = p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ˜ v − ˜ u (cid:1) p − (cid:0) ˜ v + ˜ u (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ . (34) N SPECTRAL STABILITY OF SOLER STANDING WAVES 27
We have (cid:0) ˜ v − ˜ u (cid:1) p − (cid:0) ˜ v + ˜ u (cid:1) = (cid:0) ˜ v − ˜ u (cid:1) p ˜ v + ˜ u ˜ v − ˜ u = 2 m ( p + 1) ν ν F, where F is the even function F := 1 − tanh − ν tanh × ν tanh − ν tanh . Its derivative satisfies (cid:0) − ν tanh (cid:1) F ′ = 2 (cid:0) ν − − ν (3 − ν ) tanh (cid:1) (cid:0) − tanh (cid:1) tanh . Therefore, on R + , F ′ ( t ) > ⇔ ν − > ν (3 − ν ) tanh ( t ) ⇔ t < t ν := arctanh s ν − ν (3 − ν ) , if ν > ⇔ ω m , ∅ , if ν < ⇔ ω > m . Consequently, || F || ∞ = F (0) = 1 and || ( v − u ) p − ( v + u ) || ∞ = 2 m ( p + 1) ν ν if ν < , otherwise || F || ∞ = F ( t ν ) =
18 1+ νν ν − ν and || ( v − u ) p − ( v + u ) || ∞ = m p +12 1+ ν − ν .We therefore have proved (33). (cid:3) We end this section with some computations combining the expression for ||| Q ||| withthe bounds of the previous section. The goal is to estimate the range of ( p, ω ) forwhich eigenvalues z ∈ C \ R ∪ i R associated to H satisfy Re z > E . For the sake ofcompleteness, we give expressions for the case t = m (i.e., assuming L has no positiveeigenvalues), and for the general case t > ω . From the expressions, we will then deriveTheorem 1.7 at the end of this section, as well as Lemma 6.3 at the end of the nextsection.By Lemma 5.2 and recording that θ = µ ||| Q ω ||| t + ω ) , the condition (26) on ω becomes4 θ + ( ξ ) > θ := µ ||| Q ω ||| t + ω = µp ( p + 1) m − ωt + ω , if ω > m ,µp p + 14 m ω ( t + ω ) , if ω m . Notice that, for any ω ∈ (0 , m ), the expression in the case ω > m/ ω m/
2, with both seen as functions of ω in the whole range(0 , m ), and that they are equal only at ω = m/ θ + —defined in (27)— being strictly decreasing and ω θ ( ω, t ) strictlyincreasing, determining the ω ’s for which inf η> max α ∈ [0 , f ( α, η ) > ξ is equivalent todetermine ˜ ω such that θ (˜ ω, t ) = θ + ( ξ ). Indeed, our ω ’s are then those verifying ω > ˜ ω .Finally, t θ ( ω, t ) being (strictly) decreasing and ω t m , taking t = ω gives us asufficient condition while taking t = m gives us a necessary condition , i.e., the largestpossible range of ω ’s with the bounds that we have. These necessary conditions are θ + ( ξ ) > µp ( p + 1) m − ωm + ω ⇔ ω > µp ( p + 1) − θ + ( ξ ) µp ( p + 1) + 4 θ + ( ξ ) m, if ω > m , θ + ( ξ ) > µp ( p + 1)4 m ω ( m + ω ) ⇔ ω > s µp ( p + 1)4 θ + ( ξ ) − ! m , if ω m . Moreover, denoting the unique p > m/ p ∗ := q µ θ + ( ξ ) − p ∗ > µp ∗ ( p ∗ + 1) = 12 θ + ( ξ )—, we have equivalently ω > ω ∗ := µp ( p + 1) − θ + ( ξ ) µp ( p + 1) + 4 θ + ( ξ ) m, if p > p ∗ , s µp ( p + 1)4 θ + ( ξ ) − ! m , if p p ∗ . (35)Similarly, the sufficient condition is θ + ( ξ ) > µp ( p + 1) m − ω ω ⇔ ω > µp ( p + 1) µp ( p + 1) + 8 θ + ( ξ ) m, if ω > m , θ + ( ξ ) > µ p ( p + 1)4 m ω ⇔ ω > s µp ( p + 1)8 θ + ( ξ ) m , if ω m , and denoting the unique p > m/ p ◦ := q µ θ + ( ξ ) − p ◦ > µp ◦ ( p ◦ + 1) = 8 θ + ( ξ )—, we have equivalently ω > ω ◦ := µp ( p + 1) µp ( p + 1) + 8 θ + ( ξ ) m, if p > p ◦ , s µp ( p + 1)8 θ + ( ξ ) m , if p p ◦ . (36)We can now choose E = 0, hence ξ = 0, in order to finish the proof of Theorem 1.7.Note that the condition ω > E/ Proof of Theorem 1.7.
Inserting ξ = 0 in (35) or (36), we have θ + (0) = √ and, taking µ = 2, we obtain p ∗ = √ √ − ∈ (1 . , .
54) and p ◦ := √ √ − ∈ (1 . , . N SPECTRAL STABILITY OF SOLER STANDING WAVES 29 necessary condition (35) becomes ω > ω ∗ := p ( p + 1) − √ p ( p + 1) + 3 √ m, if p > p ∗ , s √ p ( p + 1) − ! m , if p p ∗ , and the sufficient condition (36) becomes ω > ω ◦ := β ( p ) m , where β ( p ) = 2 p ( p + 1)2 p ( p + 1) + 3 √ , if p > p ◦ ,β ( p ) = s p ( p + 1)6 √ , if p p ◦ . (37) (cid:3) Figure 2.
In dark blue, the range ω/m > β ( p ), which is a sufficientcondition to have Re z > z of H outside the imaginaryaxis), independent of the spectrum of L . In light blue, the range of ω/m for which the same condition holds under the additional assumption that L has no positive eigenvalues. The values p ◦ < p ∗ are represented by thevertical dashed lines, and ω/m = 1 / The massive Gross–Neveu model
In the case f ( s ) = s , we can actually prove that L has no other eigenvalues than − ω and 0. To do so, we need the following result on the resonances (i.e., in this context,solutions of the eigenvalue equation in L ∞ ( R ) but not L ( R )) at − m − ω and m − ω .The formulae for these resonances that appear in [1, Lemma 5.5] seem to contain atypo in the expression of the second component S . Since we were not able to locate aderivation of the expression in the literature, we include the details. Lemma 6.1.
Let f ( s ) = s and < ω < m . Then the values m − ω and − m − ω areresonances of L with respective generalized eigenfunctions ( R, S ) T and ( S, R ) T where R := uvv − u and S := − ν − ν v − ν − u v − u . Proof.
We will equivalently prove the result for the rescaled problem defined in Sec-tion 5.1. Namely, that ˜ L (cid:18) ˜ R ˜ S (cid:19) = ( m − ω ) (cid:18) ˜ R ˜ S (cid:19) . The resonance at − m − ω is obtained by symmetry of the spectrum of L w.r.t. − ω (seeProposition 2.2), with corresponding (generalized) eigenfuntion obtained by exchangingthe spinors.We rewrite˜ R = √ ν tanh1 − ν tanh and ˜ S = − ν − ν − tanh − ν tanh = 11 − ν tanh − − ν . Then we compute˜ M = m − f (˜ v − ˜ u ) = m − m − ω ) 1 − tanh − ν tanh = m − m ν ν − tanh − ν tanh , ˜ R ′ = √ ν − tanh (cid:0) − ν tanh (cid:1) (cid:0) ν tanh (cid:1) = − mκ − ν ν ν tanh − ν tanh ˜ S and ˜ S ′ = (cid:18) − ν tanh (cid:19) ′ = 2 ν − tanh (cid:0) − ν tanh (cid:1) tanh = 4 mκ ν ν − tanh − ν tanh ˜ R , in order to conclude that (cid:16) ˜ L + ω (cid:17) (cid:18) ˜ R ˜ S (cid:19) = κ (cid:18) ˜ S ′ − ˜ R ′ (cid:19) + M (cid:18) ˜ R − ˜ S (cid:19) = m ν ν − tanh − ν tanh R − ν ν ν tanh − ν tanh ˜ S ! + m (cid:18) − ν ν − tanh − ν tanh (cid:19) (cid:18) ˜ R − ˜ S (cid:19) = m (cid:18) ˜ R ˜ S (cid:19) , where the last equality for the lower spinor is due to2(1 − ν )(1 + ν tanh ) + 4 ν (1 − tanh ) = 2(1 + ν )(1 − ν tanh ) . This concludes the proof. (cid:3)
We can now prove that L has no other eigenvalues than − ω and 0. Lemma 6.2 (Spectrum of L for f ( s ) = s ) . Let f ( s ) = s and < ω < m . Then σ ( L ) ∩ ( − m − ω, m − ω ) = {− ω, } . Proof.
Thanks to Theorem 1.5, we are left with proving that L has no eigenvaluesin ( − m − ω ; m − ω ) \ [ − ω ; 0] and, as before, it is enough by symmetry of the spectrumto prove that there is no eigenvalues in [ − ω ; m − ω ) \ [ − ω ; 0]. To proceed, we will usethe resonances given in Lemma 6.1.As in the proof of Theorem 1.5, we define A = L + ω , which admits the resonance + m with the same generalized eigenfunctions ( R, S ) T as L , and for which we equivalentlyhave to prove that it has no eigenvalues in [0; m ) \ [0; + ω ]. After the same coordinate N SPECTRAL STABILITY OF SOLER STANDING WAVES 31 transformation U as in the proof of the aforementioned theorem, this resonance givesthe bounded solutions R ∓ S = uvv − u ± ν − ν × v − ν − u v − u = ± ν − ν × ± − ν √ ν tanh( pκ · ) − tanh ( pκ · )1 − ν tanh ( pκ · )respectively to the equations( − ∂ x + M ± M ′ − m ) f = 0 . If these solutions have a single zero then, by Sturm’s oscillation theorem and since weknow by Theorem 1.5 that the groundstate energy of − ∂ x + M ± M ′ is ω , it showsthat there are no eigenvalues of A in the interval ( ω , m ) and we are done.That the solutions have a single zero is equivalent to h ± ( y ) := 1 ± − ν √ ν y − y , appearingin the numerators, having a single zero on ( − , h − ( x ) = h + ( − x ), we study h + .The roots of h + are −√ ν an d √ ν − . Since ν ∈ (0 , h + has a single zero on ( − , (cid:3) This means, in particular, that t defined in (18) verifies t = m , and we can obtaina larger range of ω ’s in Theorem 1.7 than the one given by the general case p > t = m implies that our necessary condition developed in section 5.3 is actuallya sufficient condition too. We therefore obtain, in the case p = 1, that β (1) = q √ − ' . . Finally, with similar computations for the choice E = m − ω (for which the condition ω > E/ ω > m/ z (foreigenvalues not lying on the axes) to the outer threshold of the essential spectrum of H . Lemma 6.3.
Let ωm < , p = 1 , and z ∈ C \ ( R ∪ i R ) be an eigenvalue of H .Then | Re z | > Re z > ( m − ω ) . Remark 6.4.
Of course, this factor is not optimal withing our framework, but it isclose since numerical estimates give it in (0 . , . L has only one eigenvalue in ( − ω, Strategy of the proof of Theorem 1.8.
The goal of this section is to determinethe number of eigenvalues of L between the eigenvalues − ω and 0. By simplicityand continuity with ω of eigenvalues, this number is independent of ω ∈ (0 , m ), andtherefore, it is sufficient to compute it for ω close to m . Therefore, it is sufficient tofind lower bounds on the eigenvalues of L µ , µ >
0, as ω approaches m . We will do thisby expanding the eigenvalues of L µ in the parameter κ = √ m − ω . In this limit, theoperators converge to the free Dirac operator, with the first-order correction given by aSchr¨odinger operator with P¨oschl–Teller potential, whose eigenvalues and eigenfunctionsare known explicitly.We define, for any p > µ >
0, the real number s := q p +1 p (1 + pµ ) − , (38) which is equivalently defined as the positive real number such that s ( s + 1) = p + 1 p (1 + pµ ) . Remark 7.1.
We recall the notations ⌈ x ⌉ and ⌊ x ⌋ : for x ∈ R , ⌈ x ⌉ is the smallestinteger larger or equal to x and ⌊ x ⌋ the largest integer smaller or equal to x . That is, ⌈ x ⌉ , ⌊ x ⌋ ∈ Z such that ⌈ x ⌉ − < x ⌈ x ⌉ and ⌊ x ⌋ x < ⌊ x ⌋ + 1. Theorem 7.2 (Eigenvalues in the non-relativistic limit) . Let ω ∈ (0 , m ) , p > , µ > ,and define s as in (38) . For sufficiently small κ = √ m − ω → + , the operator L µ ( ω ) has at least ⌈ s ⌉ eigenvalues λ < . . . < λ ⌈ s ⌉ in ( − ω, m − ω ) . These eigenvalues havean expansion, as κ → + , of the form λ k = 1 − p ( s + 1 − k ) m κ + O (cid:0) κ (cid:1) , k = 1 , . . . , ⌈ s ⌉ . (39) Moreover, if there exists an ( ⌈ s ⌉ + 1) -th eigenvalue λ ⌈ s ⌉ +1 in ( − ω, m − ω ) , then it ispositive and admits the lower bound λ ⌈ s ⌉ +1 > m − ω − O (cid:0) κ (cid:1) . (40) Remark 7.3.
The order of the error term O ( κ ) is by no means optimal. With slightlymore work, one obtains corrections ∼ κ = ( m − ω ) and explicit estimates for theerrors. However, since we only need the result in the non-relativistic limit to proveTheorem 1.8, we give the coarse estimates here.Notice that the bound on λ obtained in (39) is better in the non-relativistic limitthan the bound λ > − µ ||| Q ||| = − µp ( p + 1)( m − ω ) obtained in Lemma 3.3.Also, since p s ( p, µ ) is strictly decreasing on (0 , + ∞ ), for any µ >
0, we have s ( p ) > lim q → + ∞ s ( q ) = √ µ − > , ∀ p > , µ > , and ⌈ s ⌉ > p > µ >
0. In the particular case µ = 2, it gives s > ⌈ s ⌉ >
2, which means that L has at least 3 eigenvalues for any p > µ > s ( p, µ ) = p + µ + O ( p ) when p → + , hence the number of eigenvaluesdiverges when p → + , independently of µ . Corollary 7.4.
Let ω ∈ (0 , m ) and p > . In the non-relativistic limit, the operator L ( ω ) has exactly three eigenvalues − ω = λ < λ < λ = 0 in [ − ω, , and the secondeigenvalue λ satisfies λ = − p ( p + 2)2 m κ + O (cid:0) κ (cid:1) . Moreover, • if p < , then λ exists, is positive and satisfies λ = p (2 − p )2 m κ + O (cid:0) κ (cid:1) ; • if p > and if λ exists, then it is positive and admits the lower bound λ > m − ω − O (cid:0) κ (cid:1) . N SPECTRAL STABILITY OF SOLER STANDING WAVES 33
This shows, in particular, that the first positive eigenvalue, λ , is asymptotically closeto the essential spectrum for p >
1. In the case p < λ is positive but away fromthe essential spectrum and, actually, the smaller the p the more positive eigenvalues liein (0 , m − ω ). Proof of Corollary 7.4.
Remarking that, at µ = 2, s = s ( µ = 2 , p ) = p +1 p hence ⌈ s ⌉ > ⌈ s ⌉ > ⇔ p <
1, the bounds are a direct transcription of those in Theorem 7.2.Now, we have λ ( ω ) = − ω , (cid:12)(cid:12)(cid:12)(cid:12) λ ( ω ) + p ( p + 2)2 m κ (cid:12)(cid:12)(cid:12)(cid:12) O (cid:0) κ (cid:1) and | λ ( ω ) | O (cid:0) κ (cid:1) , and the next eigenvalue λ ( ω ), if it exists, is positive. Therefore, the eigenvalue 0 canonly be λ ( ω ) and we consequently have that − ω = λ ( ω ) < λ ( ω ) < λ ( ω ) = 0 arethe only eigenvalues in [ − ω, (cid:3) With that result, we can now prove Theorem 1.8, which relies on the continuity ofthe eigenvalues of L with respect to ω . Proof of Theorem 1.8.
The eigenvalues of L ( p, ω ) are simple, and continuous with re-spect to ω . We know that − ω and 0 are eigenvalues of L ( p, ω ) for any ω ∈ (0 , m ).Therefore, the number of eigenvalues in ( − ω,
0) is independent of ω ∈ (0 , m ). Theprevious corollary shows that this number equals one for large ω . (cid:3) Spectrum of L µ in the non-relativistic limit. We dedicate the rest of thissection to the proof of Theorem 7.2. A key tool is a minmax principle for eigenvaluesinside a gap in the essential spectrum of an operator, as shown first in [13, 16] (see [14, 15,28] for related results). This theorem gives a variational characterization of eigenvaluesof self-adjoint operators inside a gap in the essential spectrum. We will use the followingformulation of the principle.
Theorem 7.5 (Theorem 1 of [14]) . Let A be a self-adjoint operator wit domain D ( A ) , ina Hilbert space H . Suppose that Λ ± are orthogonal projections on H with Λ + + Λ − = H and such that F ± := Λ ± D ( A ) ⊂ D ( A ) . Define γ , the lower limit of the gap, as γ := sup x − ∈ F − \{ } h x − , Ax − i H || x − || H , (41) and γ ∞ , its upper limit, as γ ∞ := inf( σ ess ( A ) ∩ ( γ , + ∞ )) ∈ [ γ , + ∞ ] . Finally, for k ∈ N \ { } , the minmax levels are defined as γ k := inf V ⊂ F + dim V = k sup x ∈ ( V ⊕ F − ) \{ } h x, Ax i H || x || H . (42) If γ < + ∞ and the gap condition γ < γ (43) is satisfied, then for any k > either γ k is the k -th eigenvalue of A in ( γ , γ ∞ ) , countedwith multiplicity, or γ k = γ ∞ . In particular, γ ∞ > sup k > γ k > γ . We will apply this theorem to A = L µ , so D ( A ) = H ( R , C ). If the projectors Λ ± are chosen as the spectral projectors associated to ( −∞ , − m − ω ] and its complement,we obtain λ = − m − ω , λ ∞ = m − ω . All hypotheses are automatically satisfied, sothe minmax formula gives exactly the eigenvalues in the gap in the essential spectrum.However, the characterization is not very useful since these projectors are not knownexplicitely. The strength of the above theorem is that it still gives useful informationfor well-chosen explicit projections, adapted to the non-relativistic limit.While it would be possible to prove Theorem 7.2 by using the typical projectors onupper and lower spinor components, the projections Λ ± allow for a more streamlinedpresentation. Indeed, careful estimates show that the contribution from F − to theeigenvalues is of order κ only.We apply the minmax principle of Theorem 7.5 with A = L µ , D ( A ) = H ( R , C )and, with α = (2 m ) − , define the subspaces H + := n ( h, − αh ′ ) T (cid:12)(cid:12) h ∈ H ( R ) o and H − := n ( − αh ′ , h ) T (cid:12)(cid:12) h ∈ H ( R ) o . (44)The orthogonal projectors on these subspaces are given by the pseudo-differential oper-ators Λ + = F ∗
11 + α ξ (cid:18) − iαξiαξ α ξ (cid:19) F and Λ − = σ Λ + σ , where F denotes the Fourier transform and ξ the variable in Fourier space. It is clearfrom this expression that Λ + + Λ − = H and that F ± := Λ ± H ( R , C ) ⊂ H ( R , C ).These projections can be obtained by considering the spectral projectors on the positiveand negative spectrum for the free Dirac operator D m and keeping the first terms in aformal expansion for small ξ/m .As a final definition that will allow for a concise notation, we define for h ∈ H ( R , C ), l + ( h ) := ( h, − αh ′ ) T and l − ( h ) := ( − αh ′ , h ) T . (45)With this definition, F ± = (cid:8) l ± ( h ) (cid:12)(cid:12) h ∈ H ( R , C ) (cid:9) . Proof of Theorem 7.2.
We compute the quantities appearing in the Rayleigh quotient.First of all, recalling that α = (2 m ) − , we have h l + ( h ) , D m l + ( h ) i = (cid:28)(cid:18) h − αh ′ (cid:19) , (cid:18) mh − αh ′′ − h ′ + αmh ′ (cid:19)(cid:29) = m || l + ( h ) || + α || h ′ || , where we have used integration by parts to simplify the expressions. This identitymotivates the definition of Λ ± . Introducing the terms of L µ involving v and u , we areleft with h l + ( h ) , L µ l + ( h ) i (46)= ( m − ω ) || l + ( h ) || + α || h ′ || − Z ( v − u ) p (cid:16) | h | − α | h ′ | (cid:17) − pµ (cid:0)(cid:10) h, ( v − u ) p − v h (cid:11) + 2 Re (cid:10) αh ′ , ( v − u ) p − uvh (cid:11) − α (cid:10) h ′ , ( v − u ) p − u h ′ (cid:11)(cid:1) = ( m − ω ) || l + ( h ) || + α || h ′ || − Z ( v − u ) p n (1 + pµ ) | h | − α | h ′ | o − pµ (cid:0)(cid:10) h, ( v − u ) p − u h (cid:11) + 2 Re (cid:10) αh ′ , ( v − u ) p − uvh (cid:11) + α (cid:10) h ′ , ( v − u ) p − u h ′ (cid:11)(cid:1) N SPECTRAL STABILITY OF SOLER STANDING WAVES 35
Here, we rearranged the terms because u ≪ v in the non-relativistic limit, and the termson the second line will contribute only to the error term.Similarly, by using l − ( g ) = σ l + ( g ), we find h l − ( g ) , L µ l − ( g ) i = − ( m + ω ) || l − ( g ) || − α || g ′ || + Z ( v − u ) p n || g || − α || g ′ || o − µ h l − ( g ) , Ql − ( g ) i , where we will not need a detailed expansion of the last term.Finally, for the cross terms, we compute h l − ( g ) , D m l + ( h ) i = (cid:28)(cid:18) − αg ′ g (cid:19) , (cid:18) mh − αh ′′ − h ′ + h ′ / (cid:19)(cid:29) = α h g ′ , h ′′ i , and obtain (cid:12)(cid:12) h l − ( g ) , L µ l + ( h ) i − α h g ′ , h ′′ i (cid:12)(cid:12) (cid:0)(cid:12)(cid:12)(cid:12)(cid:12) ( v − u ) p (cid:12)(cid:12)(cid:12)(cid:12) ∞ + µ ||| Q ||| (cid:1) || l + ( h ) || || l − ( g ) || . We first estimate γ . Since Q is nonnegative, γ = sup g ∈ H h l − ( g ) , L µ l − ( g ) i|| l − ( g ) || sup g ∈ H h l − ( g ) , L l − ( g ) i|| l − ( g ) || − ( m + ω ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) v − u (cid:1) p (cid:12)(cid:12)(cid:12)(cid:12) ∞ . By taking a sequence of trial functions g R = R − / g ( · +2 RR ) for a smooth and normal-ized g , it can be shown that γ > − ( m + ω ) − α inf g ∈ H || g ′ || || l − g || = − ( m + ω ) , Combining with the bound in (31), we have − ( m + ω ) γ − ( m + ω ) + ( p + 1)( m − ω ) . (47)Therefore, from its definition, we have γ ∞ = m − ω for ω > p − p +1 m . Lower bound.
Next, we obtain a lower bound for the minmax levels γ k = inf E ⊂ H dim E = k sup ( g,h ) ∈ H × E ( g,h ) =(0 , h l + ( h ) + l − ( g ) , L µ ( l + ( h ) + l − ( g )) i|| l + ( h ) || + || l − ( g ) || > inf E ⊂ H dim E = k sup h ∈ Eh =0 h l + ( h ) , L µ l + ( h ) i|| l + ( h ) || . From (46), we find h l + ( h ) , L µ l + ( h ) i > ( m − ω ) || l + ( h ) || + α || h ′ || − (1 + pµ ) Z ( v − u ) p | h | − pµ (cid:0)(cid:12)(cid:12)(cid:12)(cid:12) ( v − u ) p − u (cid:12)(cid:12)(cid:12)(cid:12) ∞ + (cid:12)(cid:12)(cid:12)(cid:12) ( v − u ) p − uv (cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:1) || l + ( h ) || . We finally use the definition of s in (38) to obtain h l + ( h ) , L µ l + ( h ) i > ( m − ω ) || l + ( h ) || + 12 m || h ′ || − p κ m (cid:28) h, s ( s + 1)2 cosh( pκ · ) h (cid:29) − R || l + ( h ) || , where we used α = 1 / (2 m ) and have defined R := pµ (cid:0)(cid:12)(cid:12)(cid:12)(cid:12) | ˜ v − ˜ u | p − ˜ u ˜ v (cid:12)(cid:12)(cid:12)(cid:12) ∞ + (cid:12)(cid:12)(cid:12)(cid:12) | ˜ v − ˜ u | p − ˜ u (cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:1) + (1 + pµ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˜ v − ˜ u | p − ( p + 1) κ m cosh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ = O (cid:0) κ (cid:1) in view of the bounds (55)–(57) in the appendix.In order to obtain a precise bound here, we write everything in terms of the rescaledvariable ˜ x := pκx and obtain γ k > m − ω + p κ m inf E ⊂ H dim E = k sup h ∈ Eh =0 12 || h ′ || − s ( s +1)2 (cid:10) h, cosh − h (cid:11) || h || + α p κ || h ′ || − R . (48)At this point, we recognize the usual minmax formula for the eigenvalues of theSchr¨odinger operator with the P¨oschl–Teller potential V PT ( s, x ) = − s ( s + 1)2 cosh x . It is known that the spectrum of − f ′′ + V PT ( s, x ) f is the union of its essential spectrum[0; + ∞ ) and exactly ⌈ s ⌉ eigenvalues, which are E j := − ( s + 1 − j ) , j = 1 , . . . , ⌈ s ⌉ , (49)with the Legendre functions { P s +1 − js (tanh) } j =1 ,..., ⌈ s ⌉ as corresponding eigenfunctions.We define, for any integer k ∈ J , . . . , ⌈ s ⌉ K , the k -dimensional space V k := span (cid:8) P s +1 − js (tanh) (cid:9) j k . (50)That is, the eigenspace corresponding to the first k eigenvalues of this operator.Returning to (48), we find that for k ⌈ s ⌉ , the infimum is negative (use V k as trialsubspace), and therefore we may bound0 > inf E ⊂ H dim E = k sup h ∈ Eh =0 D h, − h ′′ − s ( s +1)2 cosh h E || h || + p κ m || h ′ || > inf E ⊂ H dim E = k sup h ∈ Eh =0 D h, − h ′′ − s ( s +1)2 cosh h E || h || = sup h ∈ V k \{ } D h, − h ′′ − s ( s +1)2 cosh h E || h || = E k . Inserting (49), we obtain γ k > m − ω − p κ m ( s + 1 − k ) − R . (51)In the case k = 1, we obtain γ > m − ω − p s κ m − R (52) Up to the additional term in the denominator. Also called
Ferrers functions , since − < tanh < N SPECTRAL STABILITY OF SOLER STANDING WAVES 37 and therefore, for sufficiently large values of ω , the gap condition γ > γ is satisfiedin view of (47). We conclude from Theorem 7.5 that the minmax levels γ k are theeigenvalues of L µ in ( γ , m − ω ).We now show that indeed, γ k = λ k , i.e., the minmax levels γ k are all the eigenvaluesof L µ in ( − ω, m − ω ). First, for sufficiently large ω , the lower bound (52) gives γ > − ω , so we conclude that the eigenvalue − ω lies below γ . Second, by Lemmas 3.3and 5.2 (for ω > m/ − ω, − µ ||| Q ||| ) = ( − ω, − µp ( p + 1)( m − ω )) . Since γ < − µp ( p + 1)( m − ω ) for sufficiently large ω , { γ k } k > are all the eigenvaluesof L µ in ( − ω, m − ω ).Returning to the lower bound (51), in combination with the expansion m − ω = κ m + O (cid:0) κ (cid:1) , , (53)we obtain the required lower bound for k ⌈ s ⌉ .For k > ⌈ s ⌉ , any subspace of dimension k contains a function in the positive eigenspaceof the P¨oschl–Teller Schr¨odinger operator, hence γ ⌈ s ⌉ +1 > m − ω − R . Upper bound for k ⌈ s ⌉ . In order to find an upper bound, we can not get ridof the supremum over g . However, we can choose the subspace E k in the infimum inorder to match the upper bound. So, we restrict to F k := { h ( pκx ) | h ∈ V k } , where V k ,defined in (50) is the span of the first k eigenfunctions of the P¨oschl–Teller Schr¨odingeroperator. We will not use their explicit expression (yet), but use bounds on derivativesof h ∈ F k || h ′ || pκ p s ( s + 1) || h || and || h ′′ || p κ s ( s + 1) || h || , see Appendix A.2 for details. For h ∈ V k , we bound |h l − ( g ) , L µ l + ( h ) i| = (cid:12)(cid:12) α h g ′ , h ′′ i + (cid:10) l − ( g ) , ( v − u ) p σ l + ( g ) (cid:11) − µ h l − ( g ) , Ql + ( g ) i (cid:12)(cid:12) p κ s ( s + 1) 3 m + ω m ( m + ω ) || l − ( g ) || || l + ( h ) || := R || l − ( g ) || || l + ( h ) || , where we have used (31) and Lemma 5.2 (for ω > m/ R = O ( κ ). Therefore, γ k = inf E ⊂ H dim E = k sup ( g,h ) ∈ ( H × E ) \{ } h l + ( h ) + l − ( g ) , L µ ( l + ( h ) + l − ( g )) i|| l + ( h ) || + || l − ( g ) || sup ( g,h ) ∈ ( H × F k ) \{ } h l + ( h ) , L µ l + ( h ) i + h l − ( g ) , L µ l − ( g )) i + 2 Re h l − ( g ) , L µ l + ( h ) i|| l + ( h ) || + || l − ( g ) || sup ( g,h ) ∈ ( H × F k ) \{ } h l + ( h ) , L µ l + ( h ) i + γ || l − ( g ) || + 2 R || l − ( g ) || || l + ( h ) |||| l + ( h ) || + || l − ( g ) || sup ( g,h ) ∈ ( H × F k ) \{ } h l + ( h ) , L µ l + ( h ) i + κ || l + ( h ) || + ( γ + κ − R ) || l − ( g ) || || l + ( h ) || + || l − ( g ) || , where we used the definition of γ . For the maximization over g , we note that, with r = || l − ( g ) || / || l + ( h ) || , the quotient is of the form A + Br r , If B < A , the maximum is attained at r = 0. Now, we note that A := h l + ( h ) , L µ l + ( h ) i|| l + ( h ) || + κ > m − ω − p s κ m − R by the estimates leading to (52). On the other hand, for sufficiently small κ , κ − R ∼ κ hence B := λ + κ − R < A . We finally insert (46) again, neglect the nonpositive terms, bound (out of the supre-mum) the terms of order κ or higher, insert the P¨oschl–Teller potential(1 + µp ) (cid:28) h, ( p + 1) κ m cosh h (cid:29) = p κ m (cid:28) h, s ( s + 1)2 cosh h (cid:29) in the numerator, neglect the negative term − (cid:18) | ˜ v − ˜ u | p − ( p + 1) κ m cosh (cid:19) < , see (57), then finally rescale variables in the remaining supremum, and are left with γ k sup h ∈ F k h l + ( h ) , L µ l + ( h ) i + κ || l + ( h ) || || l + ( h ) || m − ω + sup h ∈ F k α || h ′ || − (1 + µp ) R ( v − u ) p | h | (cid:0) || h || + α || h ′ || (cid:1) + R m − ω + p κ m sup h ∈ V k D h, − h ′′ − s ( s +1)2 cosh h E || h || + α p κ || h ′ || + R where we have defined R := κ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) v − u (cid:1) p (cid:12)(cid:12)(cid:12)(cid:12) ∞ κ s ( s + 1) + µp (cid:12)(cid:12)(cid:12)(cid:12) ( v − u ) p − uv (cid:12)(cid:12)(cid:12)(cid:12) ∞ = O (cid:0) κ (cid:1) . Since the numerator is negative for h ∈ V k , we also use || h || + α p κ || h ′ || || h || α p κ s ( s + 1)to obtain finally γ k m − ω − p κ m ( s + 1 − k ) α p κ s ( s + 1) + R , ∀ k ∈ { , . . . , ⌈ s ⌉} . Combined, the upper and lower bounds show that for sufficiently small κ , the minmaxlevels verify γ k ∈ ( − ω, m − ω ) for k = 1 , . . . , ⌈ s ⌉ and therefore coincide with eigenvaluesof L µ . This concludes the proof of Theorem 7.2. (cid:3) N SPECTRAL STABILITY OF SOLER STANDING WAVES 39
Appendix A. Details on some computations.
A.1. L ∞ -norms of terms involving ˜ u and ˜ v . We give here the details on computingseveral L ∞ -norms that we need in the paper. Computation of ||| ˜ v − ˜ u | p || ∞ . We have0 < | ˜ v − ˜ u | p = ( p + 1)( m − ω ) 1 − tanh − ν tanh = ( p + 1) m − ω m − ω + 2 ω cosh ( p + 1)( m − ω ) = (cid:12)(cid:12)(cid:12)(cid:12) | ˜ v − ˜ u | p (cid:12)(cid:12)(cid:12)(cid:12) ∞ = | ˜ v − ˜ u | p (0)= p + 12 m κ + p + 18 m κ + O (cid:0) κ (cid:1) . (54) Computation of ||| ˜ v − ˜ u | p − ˜ u ˜ v || ∞ . Defining h ( y ) := y ( − y ) (1 − νy ) , we have | ˜ v − ˜ u | p − ˜ u ˜ v = ( p + 1) √ ν ( m − ω ) h (tanh) . Since h , defined on ( − , ±√ y , with y := − − ν )+ √ − ν ) +4 ν ν , its ex-trema ± √ y (1 − y )(1 − νy ) , we have the non-relativistic expansion (cid:12)(cid:12)(cid:12)(cid:12) | ˜ v − ˜ u | p − ˜ u ˜ v (cid:12)(cid:12)(cid:12)(cid:12) ∞ = ( p + 1) √ ν ( m − ω ) || h || ∞ = ( p + 1)( m − ω ) √ y (1 − y )(1 − νy ) = p + 16 √ m κ (cid:0) O (cid:0) κ (cid:1)(cid:1) . (55) Computation of ||| ˜ v − ˜ u | p − ˜ u || ∞ . Since on [0 , h ( y ) := − y − νy νy − νy is nonnega-tive with a maximum ν − ν ) = m − ω ω at − ν , we have0 | v − u | p − u = ( p + 1)( m − ω ) h (cid:0) tanh (cid:1) ( p + 1) ( m − ω ) ω = (cid:12)(cid:12)(cid:12)(cid:12) | ˜ v − ˜ u | p − ˜ u (cid:12)(cid:12)(cid:12)(cid:12) ∞ = p + 132 m κ (cid:0) O (cid:0) κ (cid:1)(cid:1) . (56) Computation of (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˜ v − ˜ u | p − ( p +1) κ m cosh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ . Since h ( y ) := m − ω +2 ωy − my is positiveon R + with a maximum m √ m −√ ω √ m + √ ω at (cid:0) p mω (cid:1) , we have0 < | ˜ v − ˜ u | p − ( p + 1) κ m cosh = ( p + 1) κ h (cid:0) cosh (cid:1) ( p + 1) κ m √ m − √ ω √ m + √ ω = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˜ v − ˜ u | p − ( p + 1) κ m cosh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ = ( p + 1) κ m (cid:0) O (cid:0) κ (cid:1)(cid:1) . (57) A.2.
Bounds on derivatives of h ∈ F k . For fixed s defined in (38), we denote by p j ( x ) = P s +1 − js (tanh( pκx )) , j = 1 , . . . , ⌈ s ⌉ , the eigenfunctions of the Schr¨odinger operator − ∂ x + p κ V PT ( s, pκx ) = − ∂ x − p κ s ( s + 1)2 cosh ( pκx ) . They satisfy the eigenvalue equation − p ′′ j ( x ) = p κ (cid:18) s ( s + 1)cosh ( pκx ) − ( s + 1 − j ) (cid:19) p j ( x ) . We need bounds on the first and second derivatives of h ∈ span( p , . . . , p k ) =: F k . Firstof all, sup h ∈ F k \{ } || h ′ |||| h || = max j ∈{ , ··· ,k } (cid:12)(cid:12)(cid:12)(cid:12) p ′ j (cid:12)(cid:12)(cid:12)(cid:12) || p j || and sup h ∈ F k \{ } || h ′′ |||| h || = max j ∈{ , ··· ,k } (cid:12)(cid:12)(cid:12)(cid:12) p ′′ j (cid:12)(cid:12)(cid:12)(cid:12) || p j || . For the first bound, we multiply the eigenvalue equation by p j , integrate by parts inthe first term and bound cosh − (cid:12)(cid:12)(cid:12)(cid:12) p ′ j (cid:12)(cid:12)(cid:12)(cid:12) = p κ (cid:18)(cid:28) p j , s ( s + 1)cosh ( pκ · ) p j (cid:29) − ( s + 1 − j ) || p j || (cid:19) p κ s ( s + 1) || p j || . For the second bound, we take the norm on both sides of the eigenvalue equationand, since 0 < s + 1 − j s and 0 < ( s + 1 − j ) s < s ( s + 1) for j ∈ J , ⌈ s ⌉ K , weobtain (cid:12)(cid:12)(cid:12)(cid:12) p ′′ j (cid:12)(cid:12)(cid:12)(cid:12) || p j || p κ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s ( s + 1)cosh ( pκ · ) − ( s + 1 − j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ = p κ max (cid:8) s ( s + 1) − ( s + 1 − j ) , ( s + 1 − j ) (cid:9) , Consequently, (cid:12)(cid:12)(cid:12)(cid:12) p ′′ j (cid:12)(cid:12)(cid:12)(cid:12) || p j || p κ max (cid:8) s ( s + 1) , s (cid:9) = p κ s ( s + 1) , ∀ j ∈ J , ⌈ s ⌉ K . References [1] Gregory Berkolaiko and Andrew Comech. On spectral stability of solitary waves of nonlinear Diracequation in 1D.
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Instituto de F´ısica, Pontificia Universidad Cat´olica de Chile, Vicu˜na Mackenna4860, Santiago 7820436, Chile.
Email address : [email protected] Department of Mathematics, LMU Munich, Theresienstrasse 39, 80333 Munich, andMunich Center for Quantum Science and Technology, Schellingstr. 4, 80799 Munich,Germany
Email address : [email protected] Instituto de F´ısica, Pontificia Universidad Cat´olica de Chile, Vicu˜na Mackenna4860, Santiago 7820436, Chile.
Email address : [email protected] Departamento de Ingenier´ıa Matem´atica and Centro de Modelamiento Matem´atico(CNRS IRL 2807), Universidad de Chile, Beauchef 851, Santiago, Chile
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