Spectral Stability of Small Amplitude Solitary Waves of the Dirac Equation

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U n h self-adjoint the and | tblt fsltr waves solitary of stability ; 75,Russia 27051, ∈ ∗ τ R n o h “charge-critical the for and ) βψ (1) | φ K | R n τ ω n , ) ) | ivratadhamiltonian. and -invariant ( κ βψ and x as notations ) + e ae ( waves y τ − kona h Soler the as (known O φ ≥ i → ωt ( ω | ftebehaviour the of τ 1 otenonlinear the to ∈ | 0 . K with , H ) tteedo this of end the at ω pcrlystable spectrally for 1 / ( N R Dirac τ 0 m ω n × → K < κ < for ) , / C N 0 N , m ρ Iva38 Dirac ) (1.1) (that , , 1 we in , . rigorous analytic result on spectral stability of solitary wave solutions of the nonlinear Dirac equation (with the exception of massive Thirring model in (1+1)D [PS14, CPS16] which is completely integrable); it opens the way to the proofs of asymptotic stability in the nonlinear Dirac context. The question of stability of solitary waves is answered in many cases for the nonlinear Schrödinger, Klein–Gordon, and Korteweg–de Vries equations (see e.g. the review [Str89]). In these systems, at the points represented by solitary waves, the hamiltonian function is of finite Morse index. In simpler cases, the Morse index is equal to one, and the perturbations in the corresponding direction are prohibited by a conservation law when the Kolokolov stability condition [Kol73] is satisfied. In other words, the solitary waves could be demonstrated to correspond to conditional minimizers of the energy under the charge constraint; this results not only in spectral stability but also in orbital stability [CL82, Wei85, SS85, Wei86, GSS87]. The nature of stability of solitary wave solutions of the nonlinear Dirac equation seems completely different from this picture [Rañ83, Section V]. In particular, the hamiltonian function is not bounded from below, and is of infinite Morse index; the NLS-type approach to stability fails. As a consequence, we do not know how to prove the orbital stability but via proving the asymptotic stability first. The only known exception is the above-mentioned one-dimensional completely integrable massive Thirring model, where the orbital stability was proved by means of a coercive conservation law [PS14, CPS16] coming from higher-order integrals of motion. The first numerical evidence of spectral stability of solitary waves to the cubic nonlinear Dirac equation in (1+1)D (known as the massive Gross–Neveu model) was given in [BC12a], where the spectrum of the linearization at solitary waves was computed via the Evans function technique; no nonzero-real-part eigenvalues have been detected; this result was confirmed by numerical simulations of the dynamics in [Lak18]. A similar model in dimension 1 is given by the nonlinear coupled-mode equations; the numerical analysis of spectral stability of solitary waves in such models has been done in [BPZ98, CP06, GW08]. In the absence of spectral stability, one expects to be able to prove orbital instability, in the sense of [GSS87]; in the context of the nonlinear Schrödinger equation, such instability is proved in e.g. [KS07, GO12]. If instead a particular solitary wave is spectrally stable, one hopes to prove the asymptotic stability. Let us give a brief account on asymptotic stability results in dispersive models with unitary invariance. The asymptotic stability for the nonlinear Schrödinger equation is proved using the dispersive properties; see for instance the seminal works [SW90, SW92] for small amplitude solitary waves bifurcating from the ground state of the linear Schrödinger equation (thus with a potential) and [BP92a, BP92b, BP92c, BP95] in the translation- invariant case, in dimension 1. Under ad hoc assumptions on the spectral stability some extensions to the nonlinear Schrödinger equation in any dimension can be expected, see [Cuc01, Cuc03, Cuc09]. The analysis of the dynamics of excited states and possible relaxation to the ground state solution for small solitary waves (in any dimension) was considered in [TY02a, TY02b, TY02c, TY02d, BS03, SW04, KS06, Sch09]. These have been improved in [PW97, Wed00, GNT04, GS05, CKP06, GS06, GS07, KZ07, Miz07, Cuc08, CM08, CT09, KMz09, KZ09, Cuc11, CP14]. This path is also developed for the nonlinear Dirac equation in [Bou06, Bou08, BC12b, PS12, CT16, CPS17]. The needed dispersive properties of Dirac type models have been studied in these references and separately in [EV97, MNNO05, DF07, DF08, D’A08, BDF11, Kop11, CD13, Kop13, KT16, EG17, EGT16]. Note that the most famous class of dispersive estimates is the one of the Strichartz estimates; this class was commonly used as a major tool for well-posedness in some of the above references. We also refer, for the well-posedness problem, to the review [Pel11]. The question of the existence of stationary solutions, related to the indefiniteness of the energy, is discussed in the review [ELS08]. While the purely imaginary essential spectrum of the linearization operator is readily available via Weyl’s theorem on the essential spectrum (see [BC16] for more details and for the background on the topic), the discrete spectrum is much more delicate. Our aim in this work is to investigate the presence of eigenvalues with positive real part, which would be responsible for the linear instability of a particular solitary wave. As ω changes, such eigenvalues can bifurcate from the point spectrum on the imaginary axis or even from the essential spectrum. In [BC16], we have already shown that the bifurcations of eigenvalues from the essential spectrum into the half-planes with Re λ = 0 are only possible from the embedded thresholds at i(m + ω ) 6 ± | | (see e.g. [BPZ98]), from the collisions of eigenvalues on the imaginary axis, from the embedded eigenvalues

2 (by [BC16, Theorem 2.2], there are no embedded eigenvalues beyond the embedded thresholds), and from the point of the collision of the edges of the continuous spectrum at λ = 0 when ω = m [CGG14] and at ± λ = mi when ω = 0 [KS02]. ± Let us mention that the linear instability in the nonrelativistic limit ω / m in the “charge-supercritical” case κ > 2/n (complementary to cases which we consider in this work) follows from [CGG14]; the restrictions in that article were κ N and n 3, but they are easily removed by using the nonrelativistic asymptotics ∈ ≤ of solitary waves obtained in [BC17]. By numerics of [CMKS+16], in the case of the pure-power nonlinearity f(τ)= τ κ, κ> 2/n, the spectral instability disappears when ω (0,m) becomes sufficiently small. | | ∈ We note that quintic nonlinear Schrödinger equation in (1+1)D and the cubic one in (2+1)D are “charge critical” (all solitary waves have the same charge), and as a consequence the linearization at any solitary wave has a 4 4 Jordan block at λ = 0, resulting in dynamic instability of all solitary waves; moreover, × there is a blow-up phenomenon in the charge-critical as well as in the charge-supercritical cases; see in particular [ZSS71, ZS75, Gla77, Wei83, Mer90]. On the contrary, for the nonlinear Dirac with the critical- power nonlinearity, the charge of solitary waves is no longer constant: by [BC17], one has ∂ωQ(φω) < 0 for ω / m, where Q(φω) is the corresponding charge ((2.14) below). As a consequence, the linearization at solitary waves in the nonrelativistic limit has no 4 4 Jordan block, which resolves into 2 2 Jordan block × × (corresponding to the unitary invariance) and two purely imaginary eigenvalues. Here is the plan of the present work. The results are stated in Section 2. In Section 3, we introduce the linearization operator and derive some technical results. In Sections 5 and 5.2, we study bifurcations of eigenvalues from the embedded thresholds at (m + ω )i in the nonrelativistic limit ω m 0. In ± | | → − particular, we develop the theory of characteristic roots of operator-valued holomorphic functions, in the spirit of [Kel51, Kel71, MS70, GS71] (this is done in Section 5.2). The bifurcations of eigenvalues from the origin are analyzed in Section 4. In Appendix A, we construct the analytic continuation of the resolvent of the free Laplace operator, ex- tending the three-dimensional approach of [Rau78] to all dimensions n 1. In Appendix B we give details ≥ on the spectral theory for the nonlinear Schrödinger equation linearized at a solitary wave. The limiting absorption principle for a particular Schrödinger operator near the threshold point is covered in Appendix C. In Appendix D, we state the Schur complement in the form which we need in the proofs. Notations. We denote N = 1, 2,... and N = 0 N. For ρ> 0, an open disc of radius ρ in the complex { } 0 { }∪ plane centered at z C is denoted by D (z ) = z C ; z z < ρ ; we also denote D = D (0). We 0 ∈ ρ 0 { ∈ | − 0| } ρ ρ denote r = x for x Rn, n N, and, abusing notations, we will also denote the operator of multiplication | | ∈ ∈ with x and x = (1+ x 2)1/2 by r and r , respectively. | | h i | | h i For s, α R, we define the weighted Sobolev spaces of CN -valued functions by ∈ α n N ′ n N s α H (R , C )= u S (R , C ), u α < , u α = r i u 2 . s ∈ k kHs ∞ k kHs kh i h− ∇i kL 2 Rn CN α Rn CN α Rn CN α Rn CN We write Ls( , ) for Hs ( , ) with α = 0 and H ( , ) for Hs ( , ) with s = 0. For 2 n N u L (R , C ), we denote u = u 2 . ∈ k k k kL For any pair of normed vector spaces E and F , let B(E, F ) denote the set of bounded linear operators from E to F . For an unbounded linear operator A acting in a Banach space X with a dense domain D(A) ⊂ X, the spectrum σ(A) is the set of values λ C such that the operator A λ : D(A) X does not have a ∈ − → bounded inverse. The null space (the kernel) of A is defined by

ker(A)= v D(A) ; Av = 0 . { ∈ } The generalized null space of A is deﬁned by

L(A) := ker(Ak)= v D(A) ; Ajv D(A) j

n ∂ D = iα + βm = i αi + βm, m> 0, m − ·∇x − ∂xi Xi=1 and the massless Dirac operator by D = iα . Above, αi and β are self-adjoint N N Dirac matrices 0 − ·∇x × which satisfy (αi)2 = β2 = I , αiαj + αjαi = 2δ I , αiβ + βαi = 0, 1 i, j n, so that D2 = N ij N ≤ ≤ m ( ∆+ m2)I . Here I denotes the N N identity matrix. The anticommutation relations lead to e.g. − N N × Tr αi = Tr β−1αiβ = Tr αi = 0, 1 i n, and similarly Tr β = 0; together with σ(αi) = σ(β) = − ≤ ≤ 1 , this yields the conclusion that N is even. Let us mention that the Clifford algebra representation theory {± } (see e.g. [Fed96, Chapter 1, 5.3]) shows that there is a relation N = 2[(n+1)/2]M, M N. Without loss § I 0 ∈ of generality, we may assume that β = N/2 . Then the anticommutation relations αi, β = 0, 0 I { } − N/2 0 σ∗ 1 i n, show that the matrices αi are block-antidiagonal, αi = i , 1 i n, where the matrices ≤ ≤ σi 0 ≤ ≤ σ , 1 i n, satisfy σ∗σ + σ∗σ = 2δ , σ σ∗ + σ σ∗ = 2δ ,1 i, j n; the Dirac operator is thus i ≤ ≤ i j j i ij i j j i ij ≤ ≤ given by

n σ∗ 0 i IN/2 0 Dm = iα x + βm = i σ ∂xi + m , m> 0. (1.2) − ·∇ − i 0 0 IN/2 Xi=1 − Acknowledgment. We are indebted to the anonymous referee for corrections and suggestions. Support from the grant ANR-10-BLAN-0101 of the French Ministry of Research is gratefully acknowledged by the first author. During the work on the project, the second author was partially supported by UniversitéBourgogne Franche–Comtéand by the Metchnikov fellowship from the French Embassy. Both authors acknowledge the financial support of the région Bourgogne Franche–Comté.

2 Main results

We consider the nonlinear Dirac equation (1.1),

i∂ ψ = D ψ f(ψ∗βψ)βψ, ψ(t,x) CN , x Rn, t m − ∈ ∈ with the Dirac operator Dm of the form (1.2). Assumption 2.1. One has f C1(R 0 ) C(R), and there are κ> 0 and K > κ such that ∈ \ { } ∩ f(τ) τ κ = O( τ K ), τf ′(τ) κ τ κ = O( τ K ); τ 1. | − | | | | | | − | | | | | | |≤ If n 3, we additionally assume that κ< 2/(n 2). ≥ − In the nonrelativistic limit ω / m, the solitary waves to nonlinear Dirac equation could be obtained as −iωt bifurcations from the solitary wave solutions ϕω(y)e to the nonlinear Schr¨odinger equation 1 iψ˙ = ∆ψ ψ 2κψ, ψ(t,y) C, y Rn. (2.1) −2m − | | ∈ ∈ By [Str77, BL83] and [BGK83] (for the two-dimensional case), the stationary nonlinear Schr¨odinger equation 1 1 u = ∆u u 2κu, u(y) R, y Rn, n 1 (2.2) −2m −2m − | | ∈ ∈ ≥

4 has a strictly positive spherically symmetric exponentially decaying solution

u C2(Rn) H1(Rn) (2.3) κ ∈ ∩ (called the ground state) if and only if 0 <κ< 2/(n 2) (any κ> 0 if n 2). Moreover, − ≤ u Hs(Rn), s 0) such that

c y −(n−1)/2e−|y| u (y) C y −(n−1)/2e−|y|, y Rn; (2.5) n,κh i ≤ κ ≤ n,κh i ∀ ∈ see e.g. [BC17, Lemma 4.5]. We set

Vˆ (t) := u ( t ), Uˆ(t) := (2m)−1Vˆ ′(t), t R; (2.6) κ | | − ∈ above, a spherically symmetric function u (y), y Rn, is understood as a function of r = y . By(2.2), the κ ∈ | | functions Vˆ C2(R) and Uˆ C1(R) (which are even and odd, respectively) satisfy ∈ ∈ 1 n 1 Vˆ + ∂ Uˆ + − Uˆ = Vˆ 2κVˆ , ∂ Vˆ + 2mUˆ = 0, t R, (2.7) 2m t t | | t ∈

′ where Uˆ(t)/t at t = 0 is understood in the limit sense, limt→0 Uˆ(t)/t = Uˆ (0). In the nonrelativistic limit ω / m, the solitary wave solutions to (1.1) are obtained as bifurcations from (Vˆ , Uˆ) [BC17]; we start with summarizing their asymptotics.

Theorem 2.1. Let n N, N = 2[(n+1)/2]. Assume that the function f in (1.1) satisﬁes Assumption 2.1 ∈ with some κ, K. There is ω (m/2, m) such that for all ω (ω , m) there are solitary wave solutions 1 ∈ ∈ 1 φ (x)e−iωt to (1.1), such that φ H2(Rn, CN ) C(Rn, CN ),ω (ω , m), with ω ω ∈ ∩ ∈ 1 φ (x)∗βφ (x) φ (x) 2/2 x Rn, ω (ω , m), (2.8) ω ω ≥ | ω | ∀ ∈ ∀ ∈ 1 and 2 2 1 φ ∞ Rn CN = O (m ω ) 2κ , ω / m. k ωkL ( , ) − More explicitly,

v(r, ω)ξ φ (x)= , r = x , ξ CN/2, ξ = 1, (2.9) ω iu(r, ω) x σ ξ | | ∈ | | r · 1 1+ 1 where v(r, ω)= ǫ κ V (ǫr,ǫ), u(r, ω)= ǫ κ U(ǫr,ǫ), r 0, with ǫ = √m2 ω2, lim u(r, ǫ) = 0, and ≥ − r→0 V (t,ǫ)= Vˆ (t)+ V˜ (t,ǫ),U(t,ǫ)= Uˆ(t)+ U˜ (t,ǫ), t R, ǫ> 0, ∈ with Vˆ (t), Uˆ (t) deﬁned in (2.6). There is b < such that 0 ∞ V (t,ǫ) + U(t,ǫ) b t −(n−1)/2e−|t|, t R, ǫ (0, ǫ ). (2.10) | | | |≤ 0h i ∀ ∈ ∀ ∈ 1 V˜ (t,ǫ) There are γ > 0 and b < such that W˜ (t,ǫ)= satisﬁes 1 ∞ U˜(t,ǫ) γhti 2κ e W˜ 1 R R2 b ǫ , ǫ (0, ǫ ), (2.11) k kH ( , ) ≤ 1 ∀ ∈ 1

5 with ǫ = m2 ω2 and 1 − 1 p κ := min 1, K/κ 1 . (2.12) − There is a C1 map ω φ H1(Rn, CN ), with 7→ ω ∈ ∂ φ H1(Rn, CN ), ∂ W˜ ( ,ǫ) H1 (R) H1 (R) C1(R, R2), ω ω ∈ ǫ · ∈ even × odd ∩ 1 R 1 R 1 R where Heven( ) and Hodd( ) denote functions from H ( ) which are even and odd, respectively; γhti 2κ−1 e ∂ W˜ ( ,ǫ) 1 R R2 = O(ǫ ), ǫ (0, ǫ ), (2.13) k ǫ · kH ( , ) ∈ 1 and there is c> 0 such that 2 2 −n+ 2κ 2 2 ∂ φ n N = cǫ κ (1 + O(ǫ )), ω = m ǫ , ǫ (0, ǫ ). k ω ωkL2(R ,C ) − ∈ 1 Additionally, assume that κ, K from Assumption 2.1 satisfy eitherpκ < 2/n, or κ = 2/n, K > 4/n. Then there is ω 2/n, then there is ω 0 for all ω (ω ,m). ω ω ∈ c Above,

Q(ψ)= ψ∗(t,x)ψ(t,x) dx (2.14) Rn Z is the charge functional which is conserved due to the U(1)-invariance of the nonlinear Dirac equation (1.1). Remark 2.2. If f satisfies Assumption 2.1, then we may assume that there are c, C > 0 such that f(τ) τ κ c τ K , f(τ) (c + 1) τ κ, τ R, (2.15) | − | | |≤ | | | |≤ | | ∀ ∈ τf ′(τ) κ τ κ C τ K , τf ′(τ) (C + κ) τ κ, τ R. (2.16) | − | | |≤ | | | |≤ | | ∀ ∈ Indeed, we could achieve (2.15) and (2.16) by modifying f(τ) for τ > 1, and since the L∞-norm of the | | resulting family of solitary waves goes to zero as ω m (see Theorem 2.1), we could then take ω / m → 1 sufficiently close to m so that φ ∞ remains smaller than one for ω (ω ,m). k ωkL ∈ 1 The eigenvalues of the linearization at solitary waves φ e−iωt with ω = ω , ω m, can only accumulate ω j j → to λ = 2mi and λ = 0; see [BC16, Theorem 2.19]. We are going to relate the families of eigenvalues of ± the linearized nonlinear Dirac equation bifurcating from λ = 0 and from λ = 2mi to the eigenvalues of ± the linearization of the nonlinear Schrödinger equation at a solitary wave. Given uκ(x), a strictly positive −iωt 1 spherically symmetric exponentially decaying solution to (2.2), then uκ(x)e with ω = 2m is a solitary wave solution to the nonlinear Schrödinger equation (2.1). The linearization at this solitary− wave (see (B.2)) is given by Re ρ 0 l Re ρ ∂ = − (2.17) t Im ρ l 0 Im ρ − + (see e.g. [Kol73]), with l± defined by 1 ∆ 1 ∆ l = u2κ, l = (1 + 2κ)u2κ, D(l )= H2(Rn). (2.18) − 2m − 2m − κ + 2m − 2m − κ ± Theorem 2.2 (Bifurcations from the origin at ω = m). Let n 1. Let f C1(R 0 ) C(R) satisfy ≥ ∈ \ { } ∩ Assumption 2.1 with some values of κ, K. Let φ e−iωt, ω (ω ,m), be the family of solitary wave solutions ω ∈ 1 to (1.1) constructed in Theorem 2.1. Let ω , ω (ω ,m), be a sequence such that ω m, and assume that λ is an eigenvalue of the j j∈N j ∈ 1 j → j linearization of (1.1) at the solitary wave φ (x)e−iωj t (see Section 3) and that Re λ = 0, λ 0. Denote ωj j 6 j → λj C 2 2 1/2 N Λj = 2 , ǫj = (m ωj ) , j , ǫj ∈ − ∈ and let Λ C be an accumulation point of the sequence (Λ ) N. Then: 0 ∈ ∪ {∞} j j∈ 6 0 l− 1. Λ0 σp R. In particular, Λ0 = . ∈ l+ 0 ∩ 6 ∞ −

2. Λ0 = 0 could be an accumulation point of Λj j∈N only when κ = 2/n and ∂ωQ(φω) > 0 for ω (ω ,m), with some ω < m. If, moreover, Λ Λ = 0, then λ R for all but finitely many ∈ ∗ ∗ j → 0 j ∈ j N. ∈ In other words, as long as ∂ωQ(φω) < 0 for ω / m, there can be no linear instability due to bifurcations from the origin: there would be no eigenvalues λ of the linearization at solitary waves with ω m such j j → that Re λ = 0, λ 0. We prove Theorem 2.2 in Section 4. j 6 j → Theorem 2.3 (Bifurcations from 2mi at ω = m). Let n 1. Let f C1(R 0 ) C(R) satisfy ± ≥ ∈ \ { } ∩ Assumption 2.1 with some values of κ, K. Let φ (x)e−iωt, ω (ω ,m), be the family of solitary wave ω ∈ 1 solutions to (1.1) constructed in Theorem 2.1. Let ω , ω (ω ,m), be a sequence such that ω m and assume that λ is an eigenvalues of the j j∈N j ∈ 1 j → j linearization of (1.1) at the solitary wave φ (x)e−iωj t (see Section 3) and that λ 2mi. Denote ωj j → 2ωj + iλj C 2 2 1/2 N Zj = 2 , ǫj = (m ωj ) , j , (2.19) − ǫj ∈ − ∈ and let Z C be an accumulation point of the sequence (Z ) N. Then: 0 ∈ ∪ {∞} j j∈ 1. Z 1 σ (l ). In particular, Z = . 0 ∈ { 2m }∪ d − 0 6 ∞ 2. If the edge of the essential spectrum of l− at 1/(2m) is a regular point of the essential spectrum of l− (neither a virtual level nor an eigenvalue), then Z = 1/(2m). 0 6 3. If Z Z and Z = 0, then λ = 2ω i for all but finitely many j N. j → 0 0 j j ∈ We note that, according to the definition (2.19),

λ = i(2ω + ǫ2Z ), j N. j j j j ∈ Remark 2.3. For deﬁnitions and more details on virtual levels (also known in this context as threshold resonances and zero-energy resonances), see e.g. [JK79] (for the case n = 3), [JN01, Sections 5, 6] (for n = 1, 2), and [Yaf10, Sections 5.2 and 7.4] (for n = 1 and n 3). For the practical purposes, in the case V 0, n 2, ≥ 6≡ ≤ the threshold point is a virtual level if it corresponds to a virtual state; a virtual state, in turn, can be characterized as an L∞-solution to ( ∆+ V )Ψ = 0 which does not belong to L2 (see e.g. [JN01, Theorem 5.2 and − 1 Theorem 6.2]); for n = 3, the virtual levels can be characterized as H−s-solutions with some (and hence any) s (1/2, 3/2] [JK79, Lemma 3.2]. ∈ Remark 2.4. We do not need to study the case λ 2mi since the eigenvalues of the linearization at solitary j →− waves are symmetric with respect to real and imaginary axes; see e.g. [BC16].

In other words, as long as l− has no nonzero point spectrum and the threshold z = 1/(2m) is a regular point of the essential spectrum, there can be no nonzero-real-part eigenvalues near 2mi in the nonrelativistic ± limit ω / m. We prove Theorem 2.3 in Section 5. Let us focus on the most essential point of our work. It is of no surprise that the behaviour of eigenvalues of the linearized operator near λ = 0, in the nonrelativistic limit ω / m, follows closely the pattern which one finds in the nonlinear Schrödinger equation with the same nonlinearity; this is the content of Theorem 2.2. In its proof in Section 4, we will make this rigorous by applying the rescaling and the Schur complement to the linearization of the nonlinear Dirac equation and recovering in the nonrelativistic limit ω m the lin- → earization of the nonlinear Schrödinger equation. Consequently, the absence of eigenvalues with nonzero real part in the vicinity of λ = 0 is controlled by the Kolokolov condition ∂ωQ(φω) < 0,ω / m (see [Kol73]).

7 In other words, in the limit ω / m, the eigenvalue families λa(ω) of the nonlinear Dirac equation linearized at a solitary wave which satisfy λa(ω) 0 as ω m are merely deformations of the eigenvalue fami- NLS → → lies λa (ω) of the nonlinear Schr¨odinger equation with the same nonlinearity (linearized at corresponding solitary waves). On the other hand, by [BC16, Theorem 2.19], there could be eigenvalue families of the linearization of the nonlinear Dirac operator which satisfy lim λ (ω) = 2mi. Could these eigenvalues go off the ω→m a ± imaginary axis into the complex plane? Theorem 2.3 states that in the Soler model, under certain spectral assumptions, this scenario could be excluded. Rescaling and the Schur complement approach will show that there could be at most N/2 families of eigenvalues with nonnegative real part (with N being the number of spinor components) bifurcating from each of 2mi when ω = m; this essentially follows from Section 5.2 ± below (see Lemma 5.12 below). At the same time, the linearization at a solitary wave has eigenvalues λ = 2ωi, each of multiplicity (at least) N/2; this follows from the existence of bi-frequency solitary waves ± [BC18] in the Soler model. Namely, if there is a solitary wave solution of the form (2.9) to the nonlinear Dirac equation (1.1), then there are also bi-frequency solitary wave solutions of the form

ψ(t,x)= φ (x)e−iωt + χ (x)eiωt, ξ,η CN/2, ξ 2 η 2 = 1, (2.20) ω,ξ ω,η ∈ | | − | | where

x ∗ v(r, ω)ξ i r σ u(r, ω)η φ (x)= , χω,η = − · , (2.21) ω,ξ i x σ u(r, ω)ξ v(r, ω)η r · with v(r, ω) and u(r, ω) from (2.9). For more details and the relation to SU(1, 1) symmetry group of the Soler model, see [BC18]. The form of these bi-frequency solitary waves allows us to conclude that 2ωi are ± eigenvalues of the linearization at a solitary wave of multiplicities N/2 (see Lemma 3.4 below). Thus, we know exactly what happens to the eigenvalues which might bifurcate from 2mi: they all turn into 2ωi and ± ± stay on the imaginary axis. Remark 2.5. The spectral stability properties of bi-frequency solitary waves (2.20) can be related to the spectral stability of standard, one-frequency solitary waves (2.9); see [BC18]. As the matter of fact, since the points 2mi belong to the essential spectrum, the perturbation theory can ± not be applied immediately for the analysis of families of eigenvalues which bifurcate from 2mi. We use ± the limiting absorption principle to rewrite the eigenvalue problem in such a way that the eigenvalue no longer appears as embedded. When doing so, we ﬁnd out that the eigenvalues 2ωi become isolated solutions to ± the nonlinear eigenvalue problem, known as the characteristic roots (or, informally, nonlinear eigenvalues). To make sure that we end up with isolated nonlinear eigenvalues, we need to be able to vary the spectral parameter to both sides of the imaginary axis. To avoid the jump of the resolvent at the essential spectrum, we use the analytic continuation of the resolvent in the exponentially weighted spaces. Finally, we show that under the circumstances of the problem the isolated nonlinear eigenvalues can not bifurcate off the imaginary axis. This part is based on the theory of the characteristic roots of holomorphic operator-valued functions [Kel51, Kel71, MS70, GS71]; see also [Mar88] and [MM03, Chapter I]. Unlike in the above references, we have to deal with unbounded operators. As a result, we ﬁnd it easier to develop our own approach; see Lemma 5.13 in Section 5.2. It is of utmost importance to us that we have the explicit description of eigenvectors corresponding to 2ωi eigenvalues (see Lemma 3.4). Knowing that 2ωi are eigenvalues of ± ± the linearization operator of particular multiplicity, we will be able to conclude that there could be no other eigenvalue families starting from 2mi; in particular, no families of eigenvalues with nonzero real part. ± We use Theorems 2.3, and 2.2 to prove the spectral stability of small amplitude solitary waves.

Theorem 2.4 (Spectral stability of solitary waves of the nonlinear Dirac equation). Let n 1. Let f ≥ ∈ C1(R 0 ) C(R) satisfy Assumption 2.1, with κ, K such that either 0 <κ< 2/n, K > κ (charge- \ { } ∩ subcritical case), or κ = 2/n and K > 4/n (charge-critical case). Further, assume that σd(l−) = 0 , and { } −iωt that the threshold z = 1/(2m) of the operator l− is a regular point of its essential spectrum. Let φω(x)e ,

8 φ H2(Rn, CN ), ω / m, be the family of solitary waves constructed in Theorem 2.1. Then there is ω ∈ ω (0,m) such that for each ω (ω ,m) the corresponding solitary wave is spectrally stable. ∗ ∈ ∈ ∗ Remark 2.6. We note that, if either κ < 2/n, K > κ, or κ = 2/n, K > 4/n, then, by Theorem 2.1, for ω / m one has ∂ωQ(φω) < 0, which is formally in agreement with the Kolokolov stability criterion [Kol73].

−iωt Proof. We consider the family of solitary wave solutions φωe , ω / m, described in Theorem 2.1. Let us assume that there is a sequence ω m and a family of eigenvalues λ of the linearization at solitary waves j → j φ e−iωj t such that Re λ = 0. ωj j 6 By [BC16, Theorem 2.19], the only possible accumulation points of the sequence λ are λ = 2mi j j∈N ± and λ = 0. By Theorem 2.3, as long as σ (l ) = 0 and the threshold of l is a regular point of the d − { } − essential spectrum, λ = 2mi can not be an accumulation point of nonzero-real-part eigenvalues; it remains ± to consider the case λ λ = 0. By Theorem 2.2 (1), if Re λ = 0 and Λ is an accumulation point of the j → j 6 0 sequence Λ = λ /(m2 ω2), then j j − j

0 l− Λ0 σp R. (2.22) ∈ l+ 0 ∩ − Above, l (see (2.18)) correspond to the linearization at a solitary wave u (x)e−iωt with ω = 1/(2m) of ± κ − the nonlinear Schr¨odinger equation (2.1). For κ 2/n, the spectrum of the linearization of the corresponding ≤ NLS at a solitary wave is purely imaginary:

0 l− σp iR. l+ 0 ⊂ − We conclude from (2.22) that one could only have Λ0 = 0; by Theorem 2.2 (2), this would require that κ = 2/n and ∂ωQ(φω) > 0 for ω / m. On the other hand, as long as κ = 2/n and K > 4/n, Theorem 2.1 yields ∂ωQ(φω) < 0 for ω / m, hence Λ0 = 0 would not be possible. We conclude that there is no family of eigenvalues (λ ) N with Re λ = 0. j j∈ j 6 Remark 2.7. We can not claim the spectral stability for all subcritical values κ (0, 2/n): We need to require ∈ that σd(l−) does not have nonzero eigenvalues (this requirement is known as the “gap condition”), while for small values of κ> 0 the operator l− has a rich discrete spectrum, and potentially any of its eigenvalues could become a source of nonzero-real-part eigenvalues of linearization of the nonlinear Dirac at a small amplitude solitary wave. Such cases would require a more detailed analysis. In particular, in one spatial dimension, we only prove the spectral stability for 1 < κ 2; the critical, quintic case (κ = 2) is included, but our proof ≤ formally does not cover the cubic case κ = 1 because of the virtual level at the threshold in the spectrum of the linearization operator corresponding to the one-dimensional cubic NLS. The analytic proof of the “gap condition” for the one-dimensional case is given in [CGNT08, Theorem 3.1]. In higher dimensions, the situation is more difficult: there, we are not aware of the analytic results on the gap condition. Our numerics show that σp(l−) = 0 and the threshold 1/(2m) is a regular point of the { } n essential spectrum of l−, with l− corresponding to the nonlinear Schrödinger equation in R (thus the spectral hypotheses of Theorem 2.2 and Theorem 2.3 are satisfied) as long as

κ > κ , where κ = 1, κ 0.621, κ 0.461 . n 1 2 ≈ 3 ≈

Let us mention the related numerical results. In two dimensions, according to [CGNT08, Fig. 4], indeed l− does not have nonzero eigenvalues for κ > κ with κ 0.6. In three dimensions, according to [DS06, Fig. 2 2 ≈ 3], l− does not have nonzero eigenvalues for κ > κ3 with some κ3 < 2/3.

9 3 Technical results

3.1 The linearization operator We assume that f C1(R 0 ) C(R) satisﬁes Assumption 2.1 (recall Remark 2.2). Let φ (x)e−iωt be a ∈ \ { } ∩ ω solitary wave solution to equation (1.1) of the form (2.9), with ω (ω1,m), where ω1 (m/2,m) is from ∈ ∈ −iωt Theorem 2.1. Consider the solution to (1.1) in the form of the Ansatz ψ(t,x) = (φω(x)+ρ(t,x))e , so that ρ(t,x) CN is a small perturbation of the solitary wave. The linearization at the solitary wave φ (x)e−iωt ∈ ω (the linearized equation on ρ) is given by

i∂ ρ = (ω)ρ, (ω)= D ω f(φ∗ βφ )β 2f ′(φ∗ βφ ) Re(φ∗ β )βφ . (3.1) t L L m − − ω ω − ω ω ω · ω Remark 3.1. Even if f ′(τ) is not continuous at τ = 0, there are no singularities in (3.1) for solitary waves with ω / m constructed in Theorem 2.1: in view of the bound f ′(τ)= O( τ κ−1) (see (2.16)) and the bound | | from below φ∗ βφ φ 2/2 (see Theorem 2.1), the last term in (3.1) could be estimated by O( φ 2κ). ω ω ≥ | ω| | ω| Since (ω) is not C-linear, in order to work with C-linear operators, we introduce the following self- L adjoint matrices of size 2N 2N: × Re αi Im αi αi = − , 1 i n; Im αi Re αi ≤ ≤ (3.2) Re β Im β 0 I β = − , J = N , Im β Re β I 0 − N where the real part of a matrix is the matrix made of the real parts of its entries (and similarly for the imaginary part of a matrix). We denote

Re φω(x) R2N φω(x)= . (3.3) Im φω(x) ∈ We mention that Jα + βm is the operator which corresponds to D acting on R2N -valued functions. ·∇x m Introduce the operator

L(ω)= Jα + βm ω f(φ∗ βφ )β 2(φ∗ β )f ′(φ∗ βφ )βφ . (3.4) ·∇x − − ω ω − ω · ω ω ω We extend this operator by C-linearity from L2(Rn, R2N ) onto

2 n 2N 2 n 2N L (R , C )= L (R , C R R ), ⊗ with domain D(L(ω)) = H1(Rn, C2N ) where L(ω) is self-adjoint. The linearization at the solitary wave in (3.1) takes the form Re ρ(t,x) ∂ ρ = JL(ω)ρ, ρ(t,x)= R2N , (3.5) t Im ρ(t,x) ∈ with J from (3.2) and with L from (3.4). By Weyl’s theorem on the essential spectrum, the essential spectrum of JL(ω) is purely imaginary, with the edges at the thresholds (m ω )i; see [BC16] for more details. ± − | | There are also embedded thresholds (m + ω )i. ± | | For the reader’s convenience, we record the results on the spectral subspace of JL(ω) corresponding to the zero eigenvalue:

Lemma 3.2. Span Jφ , ∂ i φ ; 1 i n ker(JL(ω)), { ω x ω ≤ ≤ }⊂

i 1 i Span Jφ , ∂ φ , ∂ i φ , ωx Jφ α φ ; 1 i n L(JL(ω)). ω ω ω x ω ω − 2 ω ≤ ≤ ⊂

10 The proof is in [BC16]. We mention the following relations (see e.g. [BC16]):

i 1 i JLJφ = 0, JL∂ φ = Jφ ; JL∂ i φ = 0, JL ωx Jφ α φ = ∂ i φ . (3.6) ω ω ω ω x ω ω − 2 ω x ω Remark 3.3. This lemma does not give the complete characterization of the kernel of JL(ω); for example, there are also eigenvectors due to the rotational invariance and purely imaginary eigenvalues passing through λ = 0 at some particular values of ω [CMKS+16]. We also refer to the proof of Proposition 4.12 below, which gives the dimension of the generalized null space for ω / m. Lemma 3.4. The operator (ω) from (3.1) corresponding to the linearization at a (one-frequency) solitary L wave has the eigenvalue 2ω of geometric multiplicity larger than or equal to N/2, with the eigenspace − containing the subspace Span χ ; η CN/2 , with χ deﬁned in (2.21). The operator JL(ω) of the ω,η ∈ ω,η linearization at the solitary wave (see (3.5)) has eigenvalues 2ωi of geometric multiplicity larger than or ± equal to N/2. Proof. This could be concluded from the expressions for the bi-frequency solitary waves (2.20) or veriﬁed directly. Indeed, one has 2ωχ = ( iα + (m f)β ω)χ , − ω,η − ·∇x − − ω,η ∗ and then one takes into account that φω(x) βχω,η(x) = 0, so that the last term in the expression (3.1) vanishes when applied to χω,η.

3.2 Scaling, projections, and estimates on the effective potential Let

π = (1+ β)/2, π = (1 β)/2, π± = (1 iJ)/2 (3.7) P A − ∓ be the projectors corresponding to 1 σ(β) (“particle” and “antiparticle” components) and to i σ(J) ± ∈ ± ∈ (C-antilinear and C-linear). These projectors commute; we denote their compositions by

π± = π± π , π± = π± π . (3.8) P ◦ P A ◦ A With ξ CN/2, ξ = 1, from Theorem 2.1 (see (2.9)), we denote ∈ | | Re ξ 0 Ξ =   R2N , Ξ = 1. (3.9) Im ξ ∈ | |  0      For the future convenience, we introduce the orthogonal projection onto Ξ:

2N ΠΞ = Ξ Ξ, C2N End(C ). (3.10) h · i ∈ We note that, since βΞ = Ξ, one has

ΠΞ π = π ΠΞ = ΠΞ, ΠΞ π = π ΠΞ = 0. (3.11) ◦ P P ◦ ◦ A A ◦ Denote by ψ H1(Rn, C2N ) eigenfunctions which correspond to the eigenvalues λ σ (JL(ω )); j ∈ j ∈ p j thus, one has L(ω )ψ = Jα + βm ω + v(ω ) ψ = Jλ ψ , j N, (3.12) j j ·∇x − j j j − j j ∈ where the potential v(ω) is deﬁned by (cf. (3.4))

v(x,ω)ψ(x)= f(φ∗ βφ )βψ 2φ∗ βψ f ′(φ∗ βφ )βφ . (3.13) − ω ω − ω ω ω ω 11 We will use the notations y = ǫ x Rn, j ∈ where ǫ = m2 ω2, so that Jα = ǫ Jα =: ǫ D and ∆= ǫ2∆ . For v from (3.13), deﬁne the j − j ·∇x j ·∇y j 0 j y potential V(qy,ǫ) End(C2N ) by ∈ V(y,ǫ)= ǫ−2v(ǫ−1y,ω), ω = m2 ǫ2, y Rn, ǫ (0, ǫ ). (3.14) − ∈ ∈ 1 Ψ −n/2 −1 p We deﬁne j(y)= ǫj ψj(ǫj y). With V from (3.14), L(ωj) is given by

L(ω )= ǫ D + βm ω + ǫ2V(ω ), j N, (3.15) j j 0 − j j j ∈ and the relation (3.12) takes the form

ǫ D + βm ω + Jλ + ǫ2V(ω ) Ψ = 0, j N. (3.16) j 0 − j j j j j ∈ We project (3.16) onto “particle” and “antiparticle” components and onto the i spectral subspaces of J with ∓ the aid of projectors (3.7):

ǫ D π−Ψ + (m ω iλ )π−Ψ + ǫ2π−VΨ = 0, (3.17) j 0 A j − j − j P j j P j ǫ D π−Ψ (m + ω + iλ )π−Ψ + ǫ2π−VΨ = 0, (3.18) j 0 P j − j j A j j A j ǫ D π+Ψ + (m ω + iλ )π+Ψ + ǫ2π+VΨ = 0, (3.19) j 0 A j − j j P j j P j ǫ D π+Ψ (m + ω iλ )π+Ψ + ǫ2π+VΨ = 0. (3.20) j 0 P j − j − j A j j A j We need several estimates on the potential V.

Lemma 3.5. There is C > 0 such that for all y Rn and all ǫ (0, ǫ ) one has the following pointwise ∈ ∈ 1 bounds:

2κ V(y,ǫ) C2N C u (y) , k k End ( ) ≤ | κ | 2κ π V(y,ǫ) π C2N + π V(y,ǫ) π C2N Cǫ u (y) , k P ◦ ◦ Ak End ( ) k A ◦ ◦ P k End ( ) ≤ | κ | 2κ 2κ 2κ π V(y,ǫ)+ u (y) (1 + 2κΠΞ)β π C2N Cǫ u (y) , k A ◦ | κ | ◦ Ak End ( ) ≤ | κ | 2κ 2κ 2κ π V(y,ǫ)+ u (y) (1 + 2κΠΞ)β π C2N Cǫ u (y) . k P ◦ | κ | ◦ P k End ( ) ≤ | κ |

Above, uκ is the positive radially symmetric ground state of the nonlinear Schr¨odinger equation (2.2); κ = min 1, K/κ 1 > 0 was deﬁned in (2.12). −

Proof. The bound on V(y,ǫ) C2N follows from (2.15) and (2.16): k k End ( ) −2 2 2 2 ′ 2 2 −2 2κ 2κ V(y,ǫ) C2N Cǫ f(v u ) + v f (v u ) Cǫ v C u (y) , k k End ( ) ≤ | − | | − | ≤ ≤ | κ | where ω = √m2 ǫ2, ǫ (0, ǫ ), and − ∈ 1 −1 1 −1 1+ 1 v(ǫ y ,ω) CVˆ ( y )ǫ κ , u(ǫ y ,ω) CVˆ ( y )ǫ κ , | | | |≤ | | | | | |≤ | | in the notations from Theorem 2.1, with Vˆ ( y )= u (y) (see (2.6)). | | κ The bounds on π V(y,ǫ) π C2N and π V(y,ǫ) π C2N follow from k P ◦ ◦ Ak End ( ) k A ◦ ◦ P k End ( ) −2 ′ ∗ π V π C2N + π V π C2N C ǫ f (φ βφ )vu , k P ◦ ◦ Ak End ( ) k A ◦ ◦ P k End ( ) ≤ | ω ω | where f ′(φ∗ βφ ) = f ′(v2 u2) C v 2κ−2 by (2.8) and (2.16), with v, u bounded as above. | ω ω | | − |≤ | | 12 2κ Let us prove the bound on π V(y,ǫ)+ u (y) (1 + 2κΠΞ)β π C2N . For any numbers k A ◦ | κ | ◦ Ak End ( ) Vˆ > 0 and U,ˆ V˜ , U˜ R which satisfy ∈ V 1 κ ǫ U , V˜ min ,Cǫ2 Vˆ , (3.21) 1| |≤ 2 | |≤ 2 with V = Vˆ + V˜ and U = Uˆ + U˜, there are the following bounds:

2 2 2 2 2 2κ f ǫ κ (V ǫ U ) ǫ Vˆ | − − | 2 2 2 2 2 2 2 2 κ 2 2 2 2 κ 2κ 2 2κ 2κ f ǫ κ (V ǫ U ) ǫ (V ǫ U ) + ǫ (V ǫ U ) V + ǫ V Vˆ ≤ | − − − | | − − | | − | 2K/κ 2 2 2 K 2 2(κ−1) 2 2 2 2κ−1 cǫ (V ǫ U) + O(ǫ V ǫ U )+ O(ǫ Vˆ V˜ ) ≤ κ − Cǫ2+2 Vˆ 2κ, (3.22) ≤ where we used (3.21) and also applied (2.15); similarly, using (2.16),

′ 2 2 2 2 2 2 2 2κ f ǫ κ (V ǫ U ) ǫ κ V κǫ Vˆ | − − | ′ 2 2 2 2 2 2 2 2 κ−1 2 2 2 2 2 2 κ−1 2 2κ f ǫ κ (V ǫ U ) κ ǫ κ (V ǫ U ) ǫ κ V + κǫ (V ǫ U ) V Vˆ ≤ − − − | − − | 2 2 2 2 K−1 2 2 2 2 2 2 κ−1 2 2κ 2 2κ 2κ C ǫ κ (V ǫ U ) ǫ κ V + κǫ (V ǫ U ) V V + κǫ V Vˆ ≤ | − | | − − | | − | κ Cǫ2+2 Vˆ 2κ; (3.23) ≤

′ 2 2 2 2 1+ 2 2 2 2 2 κ−1 1+ 2 3 2κ f ǫ κ (V ǫ U ) ǫ κ UV C ǫ κ (V ǫ U ) ǫ κ UV Cǫ Vˆ . (3.24) | − |≤ | − | | |≤ By (2.5), (2.11), and (2.10), we may assume that ǫ1 > 0 in Theorem 2.1 is sufﬁciently small so that for ǫ (0, ǫ1) the functions V (t,ǫ), U(t,ǫ), Vˆ (t), V˜ (t,ǫ) satisfy (3.21), pointwise in t R. Then, by (3.22), ∈ κ ∈ one has ǫ−2f Vˆ 2κ Cǫ2 Vˆ 2κ; so, | − |≤ 2κ 2κ π V + Vˆ (1 + 2κΠΞ)β π C2N π V Vˆ π C2N k A ◦ ◦ Ak End ( ) ≤ k A ◦ − ◦ Ak End ( ) −2 2κ −2 ′ 2+ 2 2 2κ 2κ C ǫ f Vˆ + C ǫ f ǫ κ U Cǫ Vˆ ; ≤ | − | | |≤ in the ﬁrst inequality, we also took into account (3.11). The above is understood pointwise in y Rn; Vˆ and ∈ U are evaluated at t = y , f and f ′ are evaluated at φ∗ βφ = V 2 ǫ2U 2 and are estimated with the aid of | | ω ω − (2.15) and (2.16). 2κ The bound on π V(y,ǫ)+ u (y) (1 + 2κΠΞ)β π C2N is derived similarly. We have: k P ◦ | κ | ◦ P k End ( ) 2κ π V + Vˆ (1 + 2κΠΞ)β π C2N k P ◦ ◦ P k End ( ) −2 −2 ∗ ′ ˆ 2κ ǫ f ǫ 2(φ πP )fπP φ + V (1 + 2κΠΞ) C2N ≤k− − ω · ω k End ( ) κ ǫ−2f Vˆ 2κ + ǫ−22f ′v2 Vˆ 2κ2κ C ǫ−2f Vˆ 2κ + C ǫ−2f ′u2 Cǫ2 Vˆ 2κ. ≤ | − | | − |≤ | − | | |≤ Above, we took into account that

∗ ∗ ∗ 2 (φ βπ )π φ = ((π βφ ) )π φ = ((π φ ) )π φ = v ΠΞ, ω P · P ω P ω · P ω P ω · P ω with ΠΞ from (3.10) and v = v( x ,ω) from (2.9), since | | 1 ξ (1 + β)φ (x)= v( x ,ω) , hence π φ (x)= v( x ,ω)Ξ. 2 ω | | 0 P ω | |

13 4 Bifurcations from the origin

Here we prove Theorem 2.2.

Lemma 4.1. Let ω (0,m), j N; ω m. If there are eigenvalues λ σ JL(ω )) , Re λ = 0, j ∈ ∈ j → j ∈ p j j 6 j N, such that lim λj = 0, then the sequence ∈ j→∞

λj N Λj := 2 , j ǫj ∈ does not have the accumulation point at inﬁnity.

Proof. Due to the exponential decay of solitary waves stated in Theorem 2.1, there is C > 0 and s > 1/2 such that 2s r V( ,ω) ∞ Rn C2N C, ω (0,m). (4.1) kh i · kL ( , End ( )) ≤ ∀ ∈ Let Ψ L2(Rn, C2N ), j N, be the eigenfunctions of JL(ω ) corresponding to λ ; we then have j ∈ ∈ j j (ǫ D + βm ω + Jλ )Ψ = ǫ2V(ω )Ψ j 0 − j j j − j j j (see (3.16)). Applying to this relation π± = (1 iJ)/2 and denoting Ψ± = π±Ψ , we arrive at the system ∓ j j Ψ+ 2 + Ψ (ǫjD0 + βm ωj + iλj) j = ǫj π V(ωj) j, − − (4.2) (ǫ D + βm ω iλ )Ψ− = ǫ2π−V(ω )Ψ . j 0 − j − j j − j j j Due to ω m, without loss of generality, we can assume that ω > m/2 for all j N. Since the spectrum j → j ∈ σ(JL) is symmetric with respect to real and imaginary axes, we may assume, without loss of generality, that Im λ 0 for all j N, so that Re(iλ ) 0 (see Figure 1). Since λ 0, we can also assume that j ≥ ∈ j ≤ j → λ m/2 for all j N. With ǫ D + βm ω = D ω (considered in L2(Rn, C R R2N )) being | j| ≤ ∈ j 0 − j m − j ⊗ self-adjoint, one has 1 1 (ǫ D + βm ω iλ )−1 = = , j N. (4.3) k j 0 − j − j k dist(iλ ,σ(D ω )) m ω iλ ∀ ∈ j m − j | − j − j|

iλj −m−ωj s 0 m−ωj s −iλj

Figure 1: σ(Dm ωj ) and iλj . − ± From (4.2) and (4.3), using the bound (4.1) on V, we obtain

2 −VΨ 2 − ǫj π j ǫj −2s Ψ 2 k k C r Ψ , j N. (4.4) k j kL ≤ m ω iλ ≤ m ω iλ kh i jk ∀ ∈ | − j − j| | − j − j| From (4.2) we have: +Ψ + Ψ m ωj + iλj ǫjD0 πP j 2 πP V j − + = ǫj + , (4.5) ǫjD0 m ωj + iλj π Ψj − π VΨj − − A A hence

+ + π Ψ m + ω iλ ǫ D −1 π VΨ P j = j − j j 0 µ +∆ P j , π+Ψ ǫ D (m ω + iλ ) j y π+VΨ A j j 0 − − j j A j

14 with ∆ = D2 and y − 0 ( m ωj + iλj)(m ωj + iλj) N µj = − − 2 − , j ; (4.6) ǫj ∈ we note that Im µ = 2ǫ−2(ω + Im λ ) Re λ = 0 for all j N, so that µ +∆ is invertible indeed. j − j j j j 6 ∈ j y We may assume that inf N µ > 0, or else there would be nothing to prove: if µ 0, we would have j∈ | j| j → λ i(m ω ) = o(ǫ2), hence λ = O(ǫ2). Then, by the limiting absorption principle (cf. Lemma A.3), | j − − j | j | j| j r −sπ+Ψ C µ −1/2 r sπ+VΨ + Cǫ r sπ+VΨ . kh i jk≤ | j| kh i jk jkh i jk The above, together with (4.4) and the bound (4.1) on V, leads to

ǫ2 −sΨ −sΨ− −sΨ+ j 1 −sΨ r j r j + r j C + 1 + ǫj r j . kh i k ≤ kh i k kh i k≤ m ωj iλj µ 2 kh i k | − − | | j| If we had λ /ǫ2 , then m ω iλ λ /2 for j large enough, hence | j| j →∞ | − j − j| ≥ | j| µ m m ω iλ /ǫ2 m λ /(2ǫ2) | j|≥ | − j − j| j ≥ | j| j for j large enough (since ω m and λ 0 in (4.6)), j → j → ǫ2 −sΨ j ǫj −sΨ r j C + 1 + ǫj r j . kh i k≤ λj λ 2 kh i k | | | j| 2 Ψ N Due to λj /ǫj , the above relation would lead to a contradiction since j 0 for all j . We | | → ∞ 2 6≡ ∈ conclude that the sequence Λj = λj/ǫj can not have an accumulation point at inﬁnity.

Lemma 4.2. For any η C R there is s (η) (0, 1), lower semicontinuous in η, such that the resolvent ∈ \ + 0 ∈ ( ∆ η)−1 deﬁnes a continuous mapping − − ( ∆ η)−1 : L2(Rn) H2(Rn), 0 s

s [ r , ∆]u C u 1 O(s) C f 2 O(s), k h i − k≤ k kH ≤ k kLs one concludes from (4.7) that ( ∆ η)( r su) L2(Rn) and hence r su L2(Rn), both being bounded − − h i ∈ h i ∈ by C f 2 , with some C = C(η) < , thus so is u 2 . k kLs ∞ k kHs It is convenient to introduce the following operator acting in L2(Rn, C2N ):

~~1 ∆ 2κ 2 n 2N K = u (1 + 2κΠΞ), D(K)= H (R , C ). (4.8) 2m − 2m − κ~~

~~Above, ΠΞ End(C2N ) is the orthogonal projector onto Ξ R2N ; see (3.9), (3.10). ∈ ∈~~

~~15 Lemma 4.3. 1.~~

0 l σ − ,N = 2; l  " + 0 # − σ(JK R )=  | (πP )  l  0 − σ σ(il ) σ( il ),N 4. l − − " + 0 # ∪ ∪ − ≥  −  The equality also holds for the point spectra.

~~2. One has:~~

~~L 2n + N, κ = 2/n; dim (JK R(π ) )= 6 (4.9) | P (2n + N + 2, κ = 2/n.~~

~~Above, K is from (4.8) and l± were introduced in (2.18).~~

Proof. We decompose L2(Rn, R(π )) into the direct sum X X , where P 1 ⊕ 2 2 n X1 = L R , Span Ξ, JΞ) , ⊥ (4.10) X = L2Rn, Span Ξ, JΞ R(π ) . 2 ∩ P Note that both Ξ and JΞ belong to R(πP ). The proof of Part 1 follows once we notice that JK is invariant l 2 Rn Ξ Ξ 0 − in the spaces X1 and X2, and that JK X is represented in L , Span , J by , while | 1 l+ 0 l − 2 Rn Ξ Ξ ⊥ R 0 − JK X is represented in L , Span , J (πP ) by I N −1 C . We also notice that | 2 ∩ 2 ⊗ l− 0 − if N = 2, then X = 0 . 2 { } The proof of Part 2 also follows from the above decomposition and the relations

~~0 l− 2n + 2, κ = 2/n; dim L(JK X ) = dim L = 6 1 l | + 0 (2n + 4, κ = 2/n − (cf. Lemma B.3) and~~

~~L L l ker l dim (JK X ) = (N 2) dim ( −) = (N 2) dim ( −)= N 2. | 2 − − −~~

0 l− 2 Rn R C Remark 4.4. JK R(π ) is represented in L , (πA) by IN/2 . | A ⊗ l− 0 − Since σ(JL) is symmetric with respect to real and imaginary axes, we assume without loss of generality that λj satisﬁes Im λ 0, j N. (4.11) j ≥ ∀ ∈ Passing to a subsequence, we assume that

~~λj C Λj = 2 Λ0 . (4.12) ǫj → ∈~~

~~Lemma 4.5. 1. If Λ σ(JK), then there is C > 0 such that 0 6∈ κ π Ψ + ǫ−1 π Ψ Cǫ2 uκΨ , j N. k P jk j k A jk≤ j k κ jk ∀ ∈~~

~~16 2. For s> 0 sufﬁciently small there is C > 0 such that~~

− −1 − κ π Ψ 1 + ǫ π Ψ 1 C u Ψ , j N. k P jkHs j k A jkHs ≤ k κ jk ∀ ∈ 3. If Λ i[1/(2m), + ), then 0 6∈ ∞ + −1 + κ π Ψ 1 + ǫ π Ψ 1 C u Ψ , j N. k P jkHs j k A jkHs ≤ k κ jk ∀ ∈ 4. If Λ i[1/(2m), + ), then 0 ∈ ∞ uκπ+Ψ + ǫ−1 uκπ+Ψ C uκΨ , j N. k κ P jk j k κ A jk≤ k κ jk ∀ ∈ 2 Proof. Let us prove Part 1. We divide (3.17), (3.19) by ǫj and (3.18), (3.20) by ǫj, arriving at

~~1 + JΛ D Ψ Ψ m+ωj j 0 πP j πP V(y,ǫj) j 2 −1 = , D m ω + ǫ JΛ ǫ πAΨj − ǫjπAV(y,ǫj)Ψj " 0 − − j j j# j which we rewrite as~~

1 u2κ(1 + 2κΠΞ)+ JΛ D π Ψ m+ωj κ j 0 P j − −1 D m ω ǫ πAΨj " 0 − − j# j π (V(y,ǫ )+ u2κ(1 + 2κΠΞ))Ψ = P j κ j . (4.13) − ǫ π (V(y,ǫ )+ JΛ )Ψ j A j j j The Schur complement of the entry m ω is given by − − j 1 2κ ∆y Tj = + ΛjJ uκ (1 + 2κΠΞ) . m + ωj − − m + ωj If Λ Λ σ(JK), then T has a bounded inverse for j sufﬁciently large, and then the operator in the j → 0 6∈ j left-hand side of (4.13) has a bounded inverse (see (D.6)). Now the proof of Part 1 follows from (4.13) once we take into account the bounds from Lemma 3.5. Let us prove Part 2. We apply π± to (4.13) and rewrite the result as

1 iΛ D π±Ψ ±V Ψ m+ωj j 0 P j πP (y,ǫj) j ± −1 ± = ± . (4.14) D m ω ǫ π Ψj − ǫjπ (V(y,ǫj) iΛj)Ψj " 0 − − j# j A A ± ± ± Denote the matrix-valued operator in the left-hand side of (4.14) by Aj . The Schur complement of (Aj )22 is given by

~~± ± ± ± −1 ± 1 ∆y Tj = (Aj )11 (Aj )12(Aj )22 (Aj )21 = iΛj . (4.15) − m + ωj ± − m + ωj~~

− 2 Since Im Λ0 0 (cf. (4.11)), Tj is invertible in L (except perhaps at ﬁnitely many values of j which ≥ − − −Ψ we disregard); writing the inverse of Aj in terms of Tj , we conclude from (4.14) that πP j H1 + −1 − k k ǫ π Ψ 1 C VΨ . Moreover, by Lemma 4.2, for sufﬁciently small s> 0, j k A jkH ≤ k jk − −1 − κ π Ψj 1 + ǫ π Ψj 1 C VΨj 2 C u Ψj , j N. k P kHs j k A kHs ≤ k kLs ≤ k κ k ∈ This proves Part 2. As long as Λ i[1/(2m), + ), Part 3 is proved in the same way as Part 2. 0 6∈ ∞ To prove Part 4, we write

~~1 + iΛ + µu2κ D π+Ψ + V 2κ Ψ m+ωj j κ 0 P j πP ( µuκ ) j −1 + = + − , (4.16) D m ω ǫ π Ψj − ǫjπ (V + iΛj)Ψj " 0 − − j# j A A ~~

~~17 with some µ 0 to be speciﬁed. The Schur complement of m ω is ≥ − − j~~

~~1 2κ ∆y Tj = + iΛj + µuκ . m + ωj − m + ωj~~

We pick µ 0 such that the threshold z = 1/(2m) is a regular point of the essential spectrum of the operator ≥ 1 + µu2κ ∆ , which is by [JN01] a generic situation; indeed, by (C.1), the virtual levels correspond to the 2m κ − 2m situation when M = 1+ µ is not invertible, with a compact operator. (For n 3, it is enough to take − K K ≥ µ = 0 since ∆ has no virtual level at z = 0; for n 2, it is enough to take µ > 0 small [Sim76].) Then, − ≤ by Lemma C.1, uκ T −1 uκ (for j large enough) is bounded in L2, and the conclusion follows from (4.16). κ ◦ j ◦ κ Above, uκ . . . uκ denotes the compositions with the operators of multiplication by uκ. κ ◦ ◦ κ κ Lemma 4.5 (1) shows that if the sequence Λj converges to a point Λ0 which were away from σ(JK), then at most ﬁnitely many of Ψj could be different from zero. Therefore, there is the inclusion

~~Λ σ(JK). 0 ∈ Proposition 4.6. If Λ = lim λ /ǫ2 and Re λ = 0 for all j N, then 0 j→∞ j j j 6 ∈ Λ σ (JK) R. 0 ∈ p ∩~~

~~Proof. From now on, we assume that the corresponding eigenfunctions Ψj (cf. (3.16)) are normalized:~~

Ψ 2 = 1, j N. (4.17) k jk ∈ By (4.13), ǫ−1D π Ψ is bounded in L2 uniformly in j N, while by Lemma 4.5 (2), (3) and (4) so is j 0 A j ∈ κ −1 Ψ κ Ψ 2 −1 Ψ uκǫj πA j. Again by (4.13), uκD0πP j is uniformly bounded in L . It follows that both ǫj πP j and −1 Ψ 1 Rn C2N ǫj πA j belong to Hloc( , ) and contain weakly convergent subsequences; we denote their limits by

Pˆ H1 (Rn, C2N ), Aˆ H1 (Rn, C2N ). (4.18) ∈ loc ∈ loc Passing to the limit in (4.13) and using the bounds from Lemma 3.5, we arrive at the following system (valid in the sense of distributions):

1 u2κ(1 + 2κΠΞ)+ JΛ D Pˆ 2m − κ 0 0 = 0. (4.19) D 2m Aˆ 0 − " # ˆ N P Let us argue that if Re λj = 0 for all j , then ˆ is not identically zero. By Lemma 4.5 (2), using the 6 ∈ "A# compactness of the Sobolev embedding H1 L2, we conclude that there is an inﬁnite subsequence (which s ⊂ we again enumerate by j N) such that ∈ π−Ψ π−Pˆ H1(Rn, C2N ), ǫ−1π−Ψ π−Aˆ H1(Rn, C2N ), j , (4.20) P j → ∈ j A j → ∈ →∞ with the strong convergence in L2. Remark 4.7. If additionally Λ i[1/(2m), + ), then T + from (4.15) is also invertible; just like above, one 0 6∈ ∞ j concludes that there is an inﬁnite subsequence (which we again enumerate by j N) such that ∈ π+Ψ π+Pˆ H1(Rn, C2N ), ǫ−1π+Ψ π+Aˆ H1(Rn, C2N ) (4.21) P j → ∈ j A j → ∈ as j , with the strong convergence in L2. →∞

18 Lemma 4.8. Let J be skew-symmetric and let L be symmetric linear operators in a Hilbert space X , with J 2 = I . If λ σ (JL) iR and Ψ X is a corresponding eigenvector, then Ψ,LΨ = 0 and − X ∈ p \ ∈ h i Ψ, JΨ = 0. h i Proof. If Re λ = 0, then both sides of the identity Ψ,LΨ = λ Ψ, JΨ equal zero since Ψ,LΨ R, 6 h i − h i h i ∈ Ψ, JΨ iR. h i∈ Lemma 4.9. If Re λ = 0 for all j N, then Pˆ− = 0. j 6 ∈ 6 Proof. Lemma 4.8 yields 0= Ψ , JΨ = i Ψ+ 2 i Ψ− 2, for all j N; thus, by (4.17), h j ji k j k − k j k ∈ Ψ+ 2 = Ψ− 2 = Ψ 2/2 = 1/2, j N. (4.22) k j k k j k k jk ∀ ∈ Therefore,

~~− 2 − 2 − 2 − 2 Pˆ = lim π Ψj = lim π Ψj + π Ψj = 1/2, j N. (4.23) k k j→∞ k P k j→∞ k P k k A k ∀ ∈ Above, in the ﬁrst two relations, we took into account (4.20).~~

~~Pˆ 2 Rn C2N C2N JK Thus, ˆ L ( , ) is not identically zero, hence Λ0 σp( ). It remains to prove that "A# ∈ × ∈~~

~~Λ R. (4.24) 0 ∈ Let us assume that, on the contrary, Λ σ (JK) (iR 0 ). By(4.11), it is enough to consider 0 ∈ p ∩ \ { }~~

~~Λ0 = ia, a > 0. (4.25)~~

By (4.23), Pˆ− 2 = 1/2. Since k k + 2 + 2 + 2 + 2 Pˆ lim π Ψj lim π Ψj + π Ψj = 1/2, k k ≤ j→∞ k P k ≤ j→∞ k P k k A k we arrive at the inequality Pˆ+ 2 Pˆ− 2 0. (4.26) k k − k k ≤ From the above and from JKPˆ = iaPˆ it follows that

Pˆ, KPˆ = Pˆ, iaJPˆ = a Pˆ+, Pˆ+ a Pˆ−, Pˆ− 0. (4.27) h i h − i h i− h i≤ Remark 4.10. If Λ belongs to the spectral gap of JK (Λ iR, Λ < 1/(2m)), then both π±Ψ and 0 0 ∈ | 0| P j ǫ−1π±Ψ (up to choosing a subsequence) converge to π±Pˆ H1(Rn, C2N ) and π±Aˆ H1(Rn, C2N ) j A j ∈ ∈ strongly in L2 (cf. Remark 4.7). Then, by the above arguments, Pˆ± 2 = 1/2 and hence Pˆ, JPˆ =0= k k h i Pˆ, KPˆ . h i l 0 − 2 n 2 Lemma 4.11. If Λ0 iR 0 , Λ0 σp , and z L (R , C ) is a corresponding eigenfunc- ∈ \ { } ∈ l+ 0 ∈ tion, then − l 0 z, + z > 0. 0 l− D E

19 0 l− Proof. Let z be an eigenfunction which corresponds to Λ0 σp iR, Λ0 = 0. Let p, q ∈ l+ 0 ∩ 6 ∈ − l 2 n p p 0 − p L (R , C) be such that z = and let Λ0 = ia with a R 0 . Then ia = results iq ∈ \ { } iq l+ 0 iq − in ap = l q and aq = l p (note that q ker(l ); otherwise one would conclude that p 0 and then also − + 6∈ − ≡ q 0, so that z 0, hence not an eigenvector). These relations lead to ≡ ≡ p, l p = a p,q = a q,p = q, ap = q, l q = q, l q , h + i h i h i h i h − i h − i hence p l+ 0 p , = p, l+p + q, l−q = 2 q, l−q > 0, iq 0 l− iq h i h i h i D E where we took into account that l is semi-positive-definite and that q ker(l ). − 6∈ − l 0 Since K is invariant in the subspaces X and X defined in (4.10), where it is represented by + 1 2 0 l − l− 0 and by a positive-definite operator IN/2−1 C , respectively (see the proof of Lemma 4.3), it follows ⊗ 0 l− from Lemma 4.11 that the quadratic form l , K = , K X + , K X = , K X + , (IN−2 −) X h · · i h · · i| 1 h · · i| 2 h · · i| 1 h · ⊗ · i| 2 is strictly positive-definite on any eigenspace of JK corresponding to Λ = ia σ (JK), a> 0. Therefore, 0 ∈ p Pˆ, KPˆ > 0. (4.28) h i The relations (4.27) and (4.28) lead to a contradiction; we conclude that (4.24) is satisfied.

~~Proposition 4.6 concludes the proof of Theorem 2.2 (1).~~

The case Λ0 = 0. Now we turn to Theorem 2.2 (2), which treats the case Λ0 = 0. Let us ﬁnd the dimension of the spectral subspace of JL(ω) corresponding to all eigenvalues which satisfy λ = o(ǫ2). | | Proposition 4.12. There is δ > 0 sufﬁciently small and ǫ > 0 such that ∂D 2 ρ(JL) for all ǫ (0, ǫ ), 2 δǫ ⊂ ∈ 2 and for the Riesz projector

1 −1 2 2 Pδ,ǫ = JL(ω) η dη, ω = m ǫ (4.29) −2πi 2 − − I|η|=δǫ p one has rank P = 2n+N if κ = 2/n, and 2n+N+2 otherwise. One also has dim ker(JL(ω)) = n+N 1. δ,ǫ 6 − Remark 4.13. Let us ﬁrst give an informal calculation of rank Pδ,ǫ, which is the dimension of the generalized null space of JL. By Lemma 3.2, due to the unitary and translational invariance, the null space is of dimension (at least) n+1, and there is (at least) a 2 2 Jordan block corresponding to each of these null vectors, resulting × in dim L(JL(ω)) 2n + 2. Moreover, the ground states of the nonlinear Dirac equation from Theorem 2.1 ≥ have additional degeneracy due to the choice of the direction ξ CN/2, ξ = 1 (cf. (2.9)). The tangent space ∈ | | to the sphere on which ξ lives is of complex dimension N/2 1. (Let us point out that the real dimension is − N 2, as it should be; we did not expect to have the real dimension N 1 since we have already factored − − out the action of the unitary group.) Thus,

~~dim L(JL(ω)) 2(n + 1) + 2 N/2 1 = 2n + N, ω / m. (4.30) ≥ −~~

Whether this is a strict inequality, depends on the Kolokolo v condition ∂ωQ(φω) = 0 which indicates the jump by 2 in size of the Jordan block corresponding to the unitary symmetry, and on the energy vanishing E(φω)= 0, which indicates jumps in size of Jordan blocks corresponding to the translation symmetry [BCS15].

~~20 Proof of Proposition 4.12. Let δ > 0 be such that~~

~~0 l− Dδ σ = 0 . ∩ l+ 0 { } − Let us deﬁne the operator~~

L (ω)= ǫ−2L(ω)= ǫ−1D + ǫ−2(βm ω)+ V(y,ω), ω (ω ,m) (4.31) 0 − ∈ 1 (cf. (3.15)), where y = ǫx, ǫ = √m2 ω2, and D is the Dirac operator in the variables y = ǫx (we recall − 0 that ǫD = ǫJα = Jα ). We rewrite (4.29) as follows: 0 ·∇y ·∇x

1 −1 2 2 Pδ,ǫ = (JL (ω) η) dη, ω = m ǫ . −2πi |η|=δ − − I p Lemma 4.14. Let 1 p = (JK η)−1π dη δ −2πi − P I|η|=δ be the Riesz projector onto the generalized null space of JK R . Then: | (πP ) 1. π P π π P π p 0 P δ,ǫ P P δ,ǫ A δ 0 as ǫ 0; πAPδ,ǫπP πAPδ,ǫπA − 0 0 L2(Rn,C4N )→L2(Rn,C4N ) → →

2. There is ǫ > 0 such that, for any ǫ (0, ǫ ), one has rank P = rank p . 2 ∈ 2 δ,ǫ δ l 0 − l l Proof. By Lemma 4.3, σ(JK) σ σ(i −) σ( i −), hence (JK η) R(π ) has a bounded ⊂ l+ 0 ∪ ∪ − − | P inverse − (JK η)−1 : H−1(Rn, R(π )) H1(Rn, R(π )) − P → P on the circle η = δ, η C, with δ > 0 sufﬁciently small (cf. Lemma B.4). | | ∈ On the direct sum (R(π )) (R(π )), the operator JL (ω) η is represented by the matrix P ⊕ A − A (ǫ, η) A (ǫ) π JL π η π JL π 11 12 := P P − P A . A (ǫ) A (ǫ, η) π JL π π JL π η 21 22 A P A A − According to (4.31),

−1 A (ǫ) 1 2 + A (ǫ) 2 −1 + A (ǫ) 1 2 = O(ǫ ), k 12 kH →L k 12 kL →H k 21 kH →L −1 ǫ2 −1 4 A22(ǫ, η) R = J + OL2→L2 (ǫ ). (4.32) | (πA) − 2m In the last equality, we used the following relation (cf. (4.31)):

~~A (ǫ, η)= π JL π η = ǫ−2(m + ω)J + π JV(y,ω)π η. 22 A A − − A A −~~

~~The Schur complement of A22(ǫ, η) (see (D.6)) is given by~~

T (ǫ, η)= A (ǫ, η) A (ǫ)A (ǫ, η)−1A (ǫ) (4.33) 11 − 12 22 21 J = π + JV η π π (ǫ−1JD + JV)π A (ǫ, η)−1π (ǫ−1JD + JV)π , P m + ω − P − P 0 A 22 A 0 P which we consider as an operator T (ǫ, η) : H1(Rn, C2N ) H−1(Rn, C2N ). With the expression (4.32) for −1 → A22(ǫ, η) , the Schur complement (4.33) takes the form

J J∆y 2 T (ǫ, η)= π + JV η + O 1 −1 (ǫ ) π . (4.34) P m + ω − − 2m H →H P 21 Using the expression (4.34), we can write the inverse of JL (ω) η, considered as a map − (JL (ω) η)−1 : L2(Rn, R(π ) R(π )) L2(Rn, R(π ) R(π )), − P ⊕ A → P ⊕ A as follows (see (D.6)):

T −1 T −1A A−1 (JL η)−1 = 12 22 . (4.35) − A−1A T −1 A−1 + A−1A T −1A A−1 − 22 21 22 22 21 12 22 Above, T = T (ǫ, η), A12 = A12(ǫ), A21 = A21(ǫ), A22 = A22(ǫ, η). Since

− T (ǫ, η) (JK η) R H1→H 1 = O(ǫ), (4.36) k − − | (πP ) k uniformly in η = δ, while JK η : H1(Rn, C2N ) H−1(Rn, C2N ) has a bounded inverse for η = δ, | | − → | | the operator T (ǫ, η) R is also invertible for η = δ as long as ǫ > 0 is sufﬁciently small, with its inverse (πP ) | −1 n | | 1 n being a bounded map from H (R , R(πP )) to H (R , R(πP )). Using (4.32), we conclude that the matrix (4.35) has all its entries, except the top left one, of order O(ǫ) (when considered in the L2 L2 operator → norm). Hence, it follows from (4.35) and (4.36) that, considering P as an operator on R(π ) R(π ), δ,ǫ P ⊕ A −1 −1 pδ 0 1 T (ǫ, η) (JK η) 0 Pδ,ǫ = − − dη + O(ǫ)= O(ǫ), − 0 0 2πi |η|=δ 0 0 I where the norms refer to L2 L 2 operator norm. This proves Lemma 4.14 (1 ). The statement (2) follows → since both Pδ,ǫ and pδ are projectors.

The statement of Proposition 4.12 on the rank of Pδ,ǫ follows from Lemma 4.14 and Lemma 4.3 (2). The dimension of the kernel of JL(ω) follows from considering the rank of the projection onto the neighborhood of the eigenvalue λ = 0 of the self-adjoint operator L :

~~1 −1 2 2 Pˆδ,ǫ = (L (ω) η) dη, ω = m ǫ , −2πi |η|=δ − − I p similarly to how it was done for Pδ,ǫ, and from the relation~~

~~ker(JL (ω)) = ker(L (ω)) = R(Pˆ ), ǫ (0, ǫ ). δ,ǫ ∈ 2 Above, δ > 0 is small enough so that~~

~~l+ 0 Dδ σ = 0 . ∩ 0 l− { } This ﬁnishes the proof of Proposition 4.12.~~

Now we return to the proof of Theorem 2.2 (2). If there is an eigenvalue family (λ ) N, λ σ (JL(ω )), j j∈ j ∈ p j N λj such that λj = 0 for all j , Λj = 2 2 0 as ωj m, then the dimension of the generalized kernel 6 ∈ m −ωj → → of the nonrelativistic limit of the rescaled system jumps up, so that dim L JL(ω) + 1 2n + N + 1, |ω

~~dim L JL(ω) + 2 2n + N + 2. |ω~~

1 Φ − κ −1 (y,ω) = ǫ φω(ǫ y), − 1 −1 e1(y,ω) = ǫ κ Jφω(ǫ y), (4.37) 2− 1 −1 e2(y,ω) = ǫ κ (∂ωφω)(ǫ y); here and below, ǫ = √m2 ω2. Noting the factor ǫ−2 in the deﬁnition of L in (4.31), we deduce from (3.6) − the relations JL (ω)e (ω) = 0, JL (ω)e (ω)= e (ω), ω (ω ,m). (4.38) 1 2 1 ∈ 1 With m θ(y)= u (y) my u (y) θ H1(Rn) (4.39) − κ κ − ·∇ κ ∈ and real-valued α, β H2(Rn) such that ∈

~~l+θ(y)= uκ(y), l−α(y)= θ(y), l+β(y)= α(y) (4.40)~~

~~(see (B.7), (B.8), and (B.10) in the proof of Lemma B.3), we deﬁne~~

~~E (y)= JΞα(y), E (y)= Ξβ(y), 3 − 4 − with Ξ R2N from (3.9), so that E , E H2(Rn, R2N ) satisfy ∈ 3 4 ∈~~

~~JKE3(y)= Ξθ(y), JKE4(y)= E3(y). (4.41)~~

Lemma 4.15. Let ω = m2 ǫ2, with ǫ from Proposition 4.12. The functions 2 − 2 2 e (ω), 1 a p2, e (ω)= P E , e (ω)= JL(ω)e (ω), ω (ω ,m), a ≤ ≤ 4 δ,ǫ 4 3 4 ∈ 2 can be extended to continuous maps e : (ω ,m] L2(Rn, R2N ), 1 a 4, with e (m) = JΞu , a 2 → ≤ ≤ 1 κ e (m)= Ξθ, and e (m) = lim e (ω)= E , 3 a 4, so that 2 a ω→m a a ≤ ≤ JKe (ω) = 0, JKe (ω)= e (ω), ω (ω ,m]; 1 2 1 ∈ 2

~~JKe3(m)= e2(m), JKe4(m)= e3(m). (4.42)~~

~~Proof. By Theorem 2.1,~~

~~1 κ − κ −1 Ξ 2 e1(y,ω)= ǫ Jφω(ǫ y)= J uκ(y)+ OH1(Rn,C2N )(ǫ ),~~

~~1 n 2N so lim e1(y,ω) is deﬁned in H (R , C ). Since ω→m~~

~~v(r, ω)= ǫ1/κ(Vˆ (ǫr)+ V˜ (ǫr,ǫ)) and u(r, ω)= ǫ1+1/κ(Uˆ(ǫr)+ U˜(ǫr,ǫ)),~~

~~1 1 ˆ ˆ ∂ǫ κ ˆ κ ˜ with V , U from (2.6), one has ∂ωv(x,ω)= ∂ω ∂ǫ ǫ V (ǫx)+ ǫ V (ǫx,ǫ) , so that~~

2− 1 −1 ǫ κ ∂ωv(ǫ y,ω) (4.43) Vˆ (y) V˜ (y,ǫ) = ω + y Vˆ (y)+ + y V˜ (y,ǫ)+ ǫ∂ V˜ (y,ǫ) . − κ ·∇ κ ·∇ ǫ 23 Using (2.11) from Theorem 2.1 to bound the y V˜ -term, one has ·∇ 2κ y V˜ ( y ,ǫ) 2 Rn = O(ǫ ); k| |∇y | | kL ( ) ˜ 2κ−1 due to (2.13) from Theorem 2.1, ∂ǫV ( ,ǫ) H1(Rn,R2) = O(ǫ ). Taking into account these estimates in (4.43), we arrive at k · k

~~2− 1 −1 1 2κ ǫ κ (∂ v)(ǫ y,ǫ)= ω Vˆ (y)+ y Vˆ (y) + O 2 Rn (ǫ ), ω − κ ·∇ L ( ) 2− 1 with a similar expression for ǫ κ ∂ωu. This leads to~~

2− 1 −1 1 2κ ǫ κ (∂ φ )(ǫ y)= ω Vˆ (y)+ y Vˆ (y) Ξ + O 2 Rn (ǫ ). (4.44) ω ω − κ ·∇ L ( ) 2− 1 −1 Taking into account that e2(y,ω)= ǫ κ (∂ωφω)(ǫ y) (cf. (4.37)), the relation (4.44) allows us to deﬁne

~~1 2− κ −1 Ξ e2(m) := lim e2(ω) = lim ǫ (∂ωφω)(ǫ )= θ, (4.45) ω→m ω→m · with θ from (4.39). By(4.44), the convergence in (4.45) is in L2(Rn, C2N ). For a = 4, one has:~~

lim e4(ω) = lim Pδ,ǫE4 = E4 + lim (Pδ,ǫ pδ)E4 = E4, ω→m ω→m ω→m − with the limit holding in L2 norm. In the last relation, we used the relation p E = E , 1 a 4, and δ a a ≤ ≤ Lemma 4.14. For a = 3, the result follows from e3(ω) = JL(ω)e4(ω) = JL(ω)Pδ,ωe4(ω) since JL(ω)Pδ,ω is a bounded operator. We also point out that not only e1(ω) and e2(ω), but also e3(ω) and e4(ω) are real-valued; this follows from the observation that E L2(Rn, R2N ) is real-valued, while P commutes with the operator K : 4 ∈ δ,ω C2N C2N of complex conjugation since JL has real coefﬁcients. → In the vector space R(Pδ,ǫ) we may choose the basis (cf. Lemma 3.2)

i ǫ i e (ω), 1 a 4; ∂ i Φ, ωy JΦ α Φ, 1 i n; Θ , 1 k N 2 , (4.46) a ≤ ≤ y − 2 ≤ ≤ k ≤ ≤ − where Φ is from (4.37) and Θ (ω) are certain vectors from ker(JL (ω)), with 1 k N 2 due to k ≤ ≤ − Proposition 4.12 (which states that rank Pδ,ǫ = 2n + N + 2, dim ker(JL (ω) )= n + N 1). |Pδ,ǫ − Remark 4.16. When n = 3 and N = 4, there are three vectors Θk(ω) corresponding to inﬁnitesimal rotations around three coordinate axes, but, as it was mentioned in [BCS15], the span of these vectors, span Θ ; 1 { k ≤ k 3 , turns out to contain the null eigenvector e (ω). ≤ } 1 R In the basis (4.46) of the space (Pδ,ǫ), the operator (JL (ω) λIN ) R is represented by − | (Pδ,ǫ) λ 1 σ (ω)00 0 0 − 1 0 λ σ (ω)00 0 0  − 2  0 0 σ (ω) λ 10 0 0 3 −  0 0 σ (ω) λ 0 0 0   4  Mω λIN =  . . . −.  , (4.47) −  . . . . λI I 0   n n   . . . . −   . . . . 0 λI 0   − n   . . . .   . . . . 0 0 λI   − N−2   24 where vertical dots denote columns of irrelevant coefﬁcients and

~~σ (ω), 1 a 4, a ≤ ≤ are some continuous functions of ω. We used Lemma 4.15 and the relations~~

i ǫ i JL ωy JΦ α Φ = ∂ i Φ, 1 i n, − 2 y ≤ ≤ which follow from (3.6), (4.31), and (4.37). Considering (4.47) at λ = 0 and ǫ = 0, one concludes from (4.42) that σ1(m)= σ3(m)= σ4(m) = 0, σ2(m) = 1. (4.48) From (4.47), we also have

det(M λ) = ( λ)2n+N (λ2 λσ (ω) σ (ω)). (4.49) ω − − − 3 − 4 Lemma 4.17. For any solitary wave φ(x)e−iωt with φ H1 (Rn) and any 1 i n, one has φ, αiφ = 0. ∈ 1/2 ≤ ≤ h i µ µ µ Proof. The local version of the charge conservation, ∂µJ = 0, with J (t,x) = ψ¯(t,x)γ ψ(t,x), when applied to a solitary wave with stationary charge and current densities, J µ(t,x) = φ¯(x)γµφ(x), yields the desired identity: for any 1 i n, ≤ ≤ n 0 i j i i 0= ∂t J (x)x dx = ∂jJ (x) x dx = J (x) dx. Rn − Rn Rn Z j=1 Z Z X Expanding JL e3(ω) over the basis in R(Pδ,ǫ) (see (4.46)), we conclude that for some continuous functions γ (ω) and ρ (ω), 1 i n, and τ (ω), 1 k N 2, there is a relation i i ≤ ≤ k ≤ ≤ − 4 n N−2 i ǫ i JL e = σ e + γ ∂ i Φ + ρ ωy JΦ α Φ + τ Θ , (4.50) 3 a a i y i − 2 k k Xa=1 Xi=1 Xk=1 valid for ω (ω2,m]. Above, σa(ω), 1 a 4, are continuous functions from (4.47). Pairing (4.50) with ∈ −1 ≤ ≤ Φ = Φ(ω)= J e1(ω), we get:

0= σ (ω) J−1e (ω), e (ω) + σ (ω) J−1e (ω), e (ω) , ω (ω ,m]. (4.51) 2 1 2 4 1 4 ∈ 2 We took into account that one has Φ, v = L e , v = e , L v = 0 for any v ker(JL ), the identities h i h 2 i h 2 i ∈ J−1e , JL e (ω) = L e , e (ω) = 0, 1 3 − h 1 3 i J− 1e , e = J−1e , JL e = L e , e = 0, h 1 3i h 1 4i −h 1 4i Φ i Φ ǫ iΦ Φ∗ Φ and also the identity ,ωy J 2 α = 0 which holds due to Lemma 4.17 and due to J 0 (the left-hand side is skew-adjointh while− all thei quantities are real-valued). Since ≡

−1 − 2 −1 −1 J e (ω), e (ω) = ǫ κ φ (ǫ ), (∂ φ )(ǫ ) h 1 2 i h ω · ω ω · i n− 2 = ǫ κ φ , ∂ φ = ∂ Q(φ )/2 (4.52) h ω ω ωi ω ω (we took into account that n 2 = 0), the relation (4.51) takes the form − κ σ (ω)∂ Q(φ )/2= µ(ω)σ (ω), ω (ω ,m], (4.53) 2 ω ω 4 ∈ 2 where µ(ω) := J−1e (ω), e (ω) is a continuous function of ω (ω ,m]. −h 1 4 i ∈ 2 Remark 4.18. By (4.52) and Lemma 4.15, ∂ Q(φ ) is a continuous function of ω (ω ,m]. ω ω ∈ 2 25 Lemma 4.19. There is ω (ω ,m) such that µ(ω) > 0 for ω <ω m. 3 ∈ 2 3 ≤ Proof. We have

~~µ(ω)= Φ, e = Φ, P (e ) = J−1e (m), e (m) + O(ǫ), −h 4i −h δ,ǫ 4 i −h 1 4 i while (4.42) yields~~

J−1e , e = Ke , e = JKe , J−1e = Ξα, KΞα > 0. −h 1 4i|ω=m −h 2 4i|ω=m −h 3 3i|ω=m Above, we used (4.41) and the explicit form of E2 and E3. Lemma 4.20. There is ω [ω ,m) such that σ (ω) 0 for ω [ω ,m]. 4 ∈ 3 3 ≡ ∈ 4 Proof. Applying (JL (ω))2 to (4.50), we get

~~3 2 2 (JL ) e3(ω)= σ3(ω)(JL ) e3(ω)+ σ4(ω)(JL ) e4(ω).~~

~~−1 Coupling this relation with J e4 and using the identities~~

J−1e , (JL )3e = e , L JL e = 0 h 4 3i h 3 3i and J−1e , (JL )2e = e , L JL e = 0 h 4 4i −h 4 4i (both of these due to skew-adjointness of L JL , taking into account that e (ω), 1 a 4, are real-valued a ≤ ≤ by Lemma 4.15, while J and L have real coefﬁcients), we have

~~σ (ω) J−1e , (JL )2e = 0. (4.54) 3 h 4 3i The factor at σ3(ω) is nonzero for ω~~

J−1e , (JL )2e = J−1e , σ e + σ JL e + σ e h 4 3i|ω=m h 4 2 1 3 3 4 3i|ω=m = J−1e , e = e , φ , h 4 1i|ω=m −h 4 i|ω=m which is positive due to Lemma 4.19. Due to continuity in ω of the coefﬁcient at σ3(ω) in (4.54), we conclude that σ (ω) is identically zero for ω [ω ,m], with some ω 0 for ω / m, then for these values of ω there are two nonzero real eigenvalues of JL (ω), one positive (indicating the linear instability) and one negative, both of magnitude ∂ Q(φ ) for ω / m; hence, there are two real eigenvalues of JL, of magnitude ∼ ω ω ǫ2 ∂ Q(φ ). This completes the proof of Theorem 2.2. ∼ ω ω p p 5 Bifurcations from embedded thresholds

~~5.1 Absence of bifurcations from the essential spectrum Now we proceed to the proof of Theorem 2.3 (1) and (2): we need to prove that the sequence (2.19),~~

2ωj + iλj C N Zj = 2 , j , (5.1) − ǫj ∈ ∈ can only accumulate to either the discrete spectrum or the threshold of the operator l− from (2.18). Moreover, we will see that the accumulation to the threshold is not possible when it is a regular point of the essential 2 spectrum of l− (neither an L -eigenvalue nor a virtual level).

~~26 Lemma 5.1. There is C > 0 such that~~

~~κ + κ u π Ψ 2 Cǫ u Ψ 2 , j N. k κ jkL ≤ jk κ jkL ∀ ∈ Proof. The relations (3.19) and (3.20) yield~~

m ω + iλ ǫ D π+Ψ π+VΨ − j j j 0 P j = ǫ2 P j , ǫ D (m + ω iλ ) π+Ψ − j π+VΨ j 0 − j − j A j A j hence + + π Ψ m + ω iλ ǫ D −1 π VΨ P j = j − j j 0 ∆ + ζ2 P j , (5.2) π+Ψ ǫ D (m ω + iλ ) y j π+VΨ A j j 0 − − j j A j with ζ2 := (ω iλ )2 m2 /ǫ2 = (8m2 + o(1))/ǫ2, Re ζ 0; j N. j j − j − j j j ≥ ∈ 2 N (Note that since ωj m, Re λj = 0, andλj 2mi, one has Im ζj = 0 for all but ﬁnitely many j which → 6 → 6 2 ∈ we discard.) The limiting absorption principle from Lemma A.3 with ν = 0, 1 and with z = ζj shows that there is C > 0 such that

uκ (∆ + ζ2)−1 uκ C ζ −1, j N, (5.3) k κ ◦ y j ◦ κk ≤ | j| ∀ ∈ uκ ǫ D (∆ + ζ2)−1 uκ Cǫ , j N, (5.4) k κ ◦ j 0 y j ◦ κk ≤ j ∀ ∈ where uκ . . . uκ denotes the compositions with the operators of multiplication by uκ. Applying (5.3) and κ ◦ ◦ κ κ (5.4) to (5.2) leads to

uκπ+Ψ C uκǫD (∆ + ζ2)−1π+VΨ + uκ(∆ + ζ2)−1π+VΨ k κ jk≤ k κ 0 y j jk k κ y j jk C uκ ǫD (∆ + ζ2)−1 uκ + uκ (∆ + ζ2)−1 uκ uκΨ ≤ k κ ◦ 0 y j ◦ κk k κ ◦ y j ◦ κk k κ jk Cǫ uκΨ , j N. ≤ jk κ jk ∀ ∈ 2κ We used the bound V(y,ǫ) C2N C u (y) from Lemma 3.5. k k End ( ) ≤ | κ | Lemma 5.2. The values Z , j N from (5.1) are uniformly bounded: There is C > 0 such that j ∈ Z C, j N. | j|≤ ∀ ∈

In other words, the sequence Zj j∈N has no accumulation point at inﬁnity. Proof. Let us consider the values of j N for which the following inequality is satisﬁed: ∈ (ω + iλ )2 m2 <ǫ2. (5.5) | j j − | j 2 2 2 This implies that (ωj + iλj) = m + O(ǫj ), hence

ω iλ = m + O(ǫ2) − j − j j (since λ 2mi as ω m); taking into account the relation ǫ2Z = 2ω iλ (see (5.1)), we arrive at j →− j → j j − j − j ǫ2Z = m ω + O(ǫ2)= O(ǫ2), j j − j j j which shows that Z are uniformly bounded for the values of j for which (5.5) takes place. Now we discard | j| these values of j N; assuming that there are infinitely many left (or else there is nothing to prove), we have: ∈ (ω + iλ )2 m2 ǫ2, j N. (5.6) | j j − |≥ j ∀ ∈ 27 We write (3.17), (3.18) as the following system: D −Ψ −VΨ m ωj iλj ǫj 0 πP j 2 πP j N − − − = ǫj − , j , (5.7) ǫjD0 (m + ωj + iλj) π Ψj − π VΨj ∈ − A A which can then be rewritten as follows: −Ψ D −1 −VΨ πP j m + ωj + iλj ǫj 0 ν πP j −Ψ = ∆y + j − Ψ , (5.8) πA j ǫjD0 (m ωj iλj) πA V j − − − with 2 2 ν (ωj + iλj) m N j := 2 − , j . (5.9) ǫj ∈ We notice that ν 1 by (5.6) and that Im ν = 0 except perhaps for finitely many values of j, which we | j| ≥ j 6 discard. Applying (5.3) and (5.4) to (5.8), we derive: uκπ−Ψ C ǫ + ν −1/2 uκΨ , j N. (5.10) k κ jk≤ j | j| k κ jk ∀ ∈ By Lemma 5.1 and (5.10), there is C > 0such that uκΨ uκπ+Ψ + uκπ−Ψ C ǫ + ν −1/2 uκΨ , j N. (5.11) k κ jk ≤ k κ jk k κ jk≤ j | j| k κ jk ∀ ∈ ν κΨ Assume that lim sup j =+ . Then the coefficient at uκ j in the right-hand side of (5.11) would go to j→∞ | | ∞ k k zero for an infinite subsequence of j ; since Ψ 0, we arrive at the contradiction. →∞ j 6≡ Thus, ν are uniformly bounded. From (5.9), we derive: | j | ω + iλ = m2 + ǫ2ν = m + O(ǫ2), j j − j j − j where we chose the “positive” branch of the squareq root (for all but finitely many j N) since ω m and ∈ j → λ 2mi as j . This yields j → →∞ 2 2 2 2ν 2ω + iλ ω + (ω + iλ ) m ǫj m + ǫj j Z = j j = j j j = − − , j N; j 2 2 q 2q − ǫj − ǫj − ǫj ∈ therefore, Zj = O(1) are uniformly bounded and could not accumulate at infinity.

m+ωj +iλj 2ωj +iλj m−ωj 1 Substituting 2 = 2 2 = Zj m+ω , we rewrite (5.7) as − ǫj − ǫj − ǫj − j D −Ψ 2 − Ψ m ωj iλj 0 πP j ǫj πP V j − − 1 2κ − = − 2κ − . (5.12) D0 Zj + uκ ǫjπ Ψj − ǫ π VΨ ǫ u π Ψ " − m+ωj # A j A j − j κ A j We denote the matrix-valued operator in the left-hand side by

m ωj iλj D0 Aj := − − 1 2κ , (5.13) D0 Zj + u " − m+ωj κ # A : L2(Rn, C4N ) L2(Rn, C4N ), D(A )= H1(Rn, C4N ); j N. j → j ∈ We note that (A ) 2m as j , hence, for j large enough, (A ) is invertible. By (D.4), the Schur j 11 → → ∞ j 11 complement of (Aj)11 is given by

−1 1 2κ ∆y N Sj = (Aj)22 (Aj)21(Aj )11 (Aj)12 = Zj + uκ + I2N , j , (5.14) − − m + ωj m ωj iλj ∈ − − with S : L2(Rn, C2N ) L2(Rn, C2N ), D(S )= H2(Rn, C2N ) for j N. j → j ∈ Let Z be the limit point of the sequence Z ; by Lemma 5.2, Z = . 0 j j∈N 0 6 ∞ 28 Lemma 5.3. Z σ (l ) 1/(2m) . If, moreover, the threshold z = 1/(2m) is a regular point of the 0 ∈ d − ∪ { } essential spectrum of l , then Z σ (l ). − 0 ∈ d − Proof. We write the inverse of the matrix-valued operator in the left-hand side of (5.12) using (D.4):

−Ψ −1 −1 −1 2 − Ψ π j 1 1 + (Aj)11 D0Sj D0 D0Sj ǫj π V j P = − P . (5.15) − −1 −1 − 2κ "ǫjπ Ψj# −(Aj)11 " S D0 (Aj)11S #"ǫjπ V u Ψj# A − j j A − κ Above, the Schur complement of (A ) = m ω iλ is given by the expression (5.14): j 11 − j − j S = h I , j N, j j 2N ∈ with h : L2(Rn) L2(Rn) (with domain D(h )= H2(Rn)) given by j → j 1 ∆ h = Z + u2κ + y j j − m + ω κ m ω iλ j − j − j 1 1 1 = Zj + m ωj iλj − m + ωj − m ωj iλj − − − 2κ− 2m 2κ 2muκ ∆y + 1 uκ + + − m ωj iλj m ωj iλj m ωj iλj − − − − − − 2m l ˆ 2m 2κ = ( − Zj)+ 1 uκ , −m ωj iλj − − m ωj iλj − − − − where the sequence

m ωj iλj 1 1 Zˆj = − − Zj + , j N, 2m m ωj iλj − m + ωj ∈ − − has the same limit as the sequence Zj j∈N. Let Ω C be a compact set as in Lemma C.1: Ω σ (l )= , and if z = 1/(2m) is an eigenvalue or a ⊂ ∩ d − ∅ virtual level of l , then we also assume that 1/(2m) Ω. Then, by Lemma C.1, there is C > 0 such that − 6∈ κ −1 κ u (l z) u 2 2 C, z Ω [1/(2m), + ). k κ ◦ − − ◦ κkL →H ≤ ∀ ∈ \ ∞ Since Z Ω, one has Zˆ Ω and also Zˆ σ(l ) for j N large enough. We note that 0 ∈ j ∈ j 6∈ − ∈ 2 ˆ m ωj iλj 2ωj + iλj 1 1 1 (m ωj iλj) Zj = − − 2 2 + = − 2 − 2 ; 2m m ωj m ω iλj − m ωj 2m − 2m(m ωj ) − − − − − ˆ (m−ωj +Im λj ) Re λj ˆ l since Im λj 2m and Re λj > 0, we conclude that Im Zj = 2 2 = 0 (hence Zj σ( −)) for → 2m(m −ωj ) 6 6∈ all j N except at most for ﬁnitely many values of j which we discard. Then the mapping ∈ uκ S−1 uκ : L2(Rn) H2(Rn), j N, κ ◦ j ◦ κ → ∈ is bounded uniformly in j N. By duality, so is the mapping H−2(Rn) L2(Rn), and, by complex ∈ → interpolation, so is uκ S−1 uκ : H−1(Rn) H1(Rn). κ ◦ j ◦ κ → It follows that the following maps are also continuous:

uκ S−1 D uκ : L2(Rn) H1(Rn), κ ◦ j ◦ 0 ◦ κ → uκ D S−1 uκ : H−1(Rn) L2(Rn), κ ◦ 0 ◦ j ◦ κ → uκ D S−1 D uκ : L2(Rn) L2(Rn), κ ◦ 0 ◦ j ◦ 0 ◦ κ → 29 with the bounds which are uniform in j N. ∈ We recall that there is the inclusion u C2(Rn) (see (2.3)), with both u and its derivatives exponentially κ ∈ κ decaying with x ([BC17, Lemma A.1]); moreover, inf Rn ∂ u(x)/u(x) > 0 ([BC17, Lemma A.2]). As | | x∈ | r | a consequence, for any κ > 0 (with κ < 2/(n 2) if n 3), the multiplication by uκ extends to a − ≥ κ continuous mapping in L2(Rn), in H1(Rn), and by duality in H−1(Rn). Similarly, the matrix multiplication κ κ−1 i 2 Rn C2N by [D0, uκ]= κuκ Jα ∂xi uκ extends to a continuous mapping in L ( , ). Therefore, if either Z Z C σ(l ) or Z Z (1/(2m), + ) (with Z = 1/(2m) if j → 0 ∈ \ − j → 0 ∈ ∞ 0 6 z = 1/(2m) either an eigenvalue or a virtual level of l ), with Im Z = 0, then, taking into account that − j 6 λ 2mi as ω m, we conclude that the relation (5.15) yields the following inequality: j → j → 1 1 κ −Ψ κ −Ψ 2 −VΨ −VΨ 2κ −Ψ uκπP j + ǫj uκπA j Cǫj κ πP j + Cǫj κ πA j uκ πA j k k k k≤ kuκ k uκ − 1 1 1 2 −VΨ −V −Ψ − V 2κ −Ψ Cǫj κ πP j + Cǫj κ πA πP j + κ πA ( uκ )πA j , (5.16) ≤ uκ uκ uκ −

~~with C = C(Z0) > 0, and then, using the bounds~~

2κ 2κ V C2N C u (y) , π V π C2N Cǫ u (y) , k k End ( ) ≤ | κ | k A ◦ ◦ P k End ( ) ≤ | κ | and − 2κ − 2κ 2κ π (V + u (1 + 2κΠΞ)β)π C2N Cǫ u (y) k A κ A k End ( ) ≤ | κ | from Lemma 3.5 and taking into account the identity

− 2κ − − 2κ − π (V u )π = π (V + u (1 + 2κΠΞ)β)π , A − κ A A κ A we obtain the estimate κ uκπ−Ψ Cǫmin(1,2 ) uκΨ , j N. (5.17) k κ jk≤ j k κ jk ∀ ∈ κ The inequality (5.17) and Lemma 5.1 lead to uκΨ = O ǫmin(1,2 ) uκΨ , in contradiction to Ψ 0, k κ jk j k κ jk j 6≡ j N. Thus, the assumption Z C σ(l ) leads to a contradiction, and so does the assumption Z ∈ 0 ∈ \ − 0 ∈ (1/(2m), + ), and so does the assumption Z = 1/(2m) when the threshold z = 1/(2m) is a regular point ∞ 0 of the essential spectrum of l−. This ﬁnishes the proof of Theorem 2.3 (1) and (2).

~~5.2 Characteristic roots of the nonlinear eigenvalue problem~~

Let us prove Theorem 2.3 (3), showing that Z0 = 0 is only possible when λj = 2ωji for all but ﬁnitely many + j N. First, we claim that the relations (3.19) and (3.20) allow one to express Y := π Ψj in terms of ∈ − X := π Ψj. Lemma 5.4. There is ǫ (0, ǫ ) such that for any ǫ (0, ǫ ) and any z D the relations 2 ∈ 1 ∈ 2 ∈ 1 ǫD π Y (ω iλ m)π Y + ǫ2π+V(X + Y) = 0, 0 A − − − P P ǫD π Y (ω iλ + m)π Y + ǫ2π+V(X + Y) = 0, 0 P − − A A where ω = √m2 ǫ2 and − λ = λ(z) = (2ω + ǫ2z)i (5.18)

(cf. (2.19)), deﬁne a linear map ϑ( ,ǫ,z) : L2,−κ(Rn, R(π−)) L2,−κ(Rn, R(π+)), ϑ( ,ǫ,z) : X Y, · → · 7→ which is analytic in z, where for µ R the exponentially weighted spaces are deﬁned by ∈ 2,µ n 2 n µhri 2 n µhri L (R )= u L (R ); e u L (R ) , u 2,µ := e u 2 . ∈ loc ∈ k kL k kL 30 Moreover, there is C > 0 such that

ϑ( ,ǫ,z) 2,−κ Rn C2N 2,−κ Rn C2N Cǫ, ǫ (0, ǫ ), z D , k · kL ( , )→L ( , ) ≤ ∈ 2 ∈ 1 3 ∂ ϑ( ,ǫ,z) 2,−κ Rn C2N 2,−κ Rn C2N Cǫ , ǫ (0, ǫ ), z D . k z · kL ( , )→L ( , ) ≤ ∈ 2 ∈ 1 Proof. By (5.2),

~~−1 π Y ω iλ + m ǫD (ω iλ)2 m2 π+V(X + Y) P = − 0 ∆ + − − I P . π Y ǫD ω iλ m y ǫ2 π+V(X + Y) A 0 − − A This leads to~~

(ω iλ)2 m2 −1 Y = π+ (ω iλ)+ m(π π )+ ǫD ∆ + − − I V(X + Y). (5.19) { − P − A 0} y ǫ2 For ǫ> 0 and z C, Im z < 0 (so that Re λ(z) > 0 by (5.18)), we deﬁne the linear map ∈ Φ( ,ǫ,z) : L2,−κ(Rn, C2N ) L2,−κ(Rn, R(π+)), (5.20) · → 2 2 −1 Φ(Ψ,ǫ,z)= π+ (ω iλ)+ m(π π )+ ǫD ∆ + (ω−iλ) −m I VΨ, { − P − A 0} y ǫ2 where λ = λ(z) = (2ω + ǫ2z)i. Using the deﬁnition (5.20), the relation (5.19) takes the form

~~Y = Φ(X + Y,ǫ,z). (5.21)~~

Since the norm of Φ( ,ǫ,z) B L2,−κ(Rn, C2N ),L2,−κ(Rn, C2N ) , z D , is small (as long as ǫ > 0 is · ∈ ∈ 1 small enough), we will be able to use the above relation to express Y as a function of X L2,−κ(Rn, R(π−)). ∈ (ω−iλ)2−m2 Remark 5.5. The inverse of ∆y + ǫ2 is not continuous in λ at Re λ = 0; this discontinuity could result in two different families of eigenvalues bifurcating from an embedded eigenvalue even when its algebraic multiplicity is one. We will use the resolvent corresponding to Re λ> 0 and then use its analytic continuation through Re λ = 0. In view of Proposition A.4, it is convenient to change the variables so that in (5.20) we deal with ( ∆ − y − ζ2); let ζ C be deﬁned by ∈ ζ2 = ((ω iλ)2 m2)/ǫ2, Re ζ 0; (5.22) − − ≥ D the values of λ with Re λ > 0 correspond to the values of ζ with Im ζ < 0 (since λ ǫ2 (2ωi) and ω ω 2 ǫ2 ǫ ǫ ∈ ω ( 2,m), where 2 = m 2, with 2 (0, 1) small enough). ∈ − ∈−2κ|y| Due to Lemma 3.5, V(y,ǫ) C2N Ce ; using the analytic continuation of the resolvent from k p k End ( ) ≤ Proposition A.4, the mapping (5.20) could be extended from Im ζ < 0 to ζ C; Im ζ < κ iR . For { } { ∈ } \ + the uniformity, we require that

Im ζ < κ, (5.23) | | considering the resolvent ( ∆ ζ2)−1 for Im ζ < 0 (this corresponds to Re λ> 0) and its analytic contin- − y − uation into the strip 0 Im ζ < κ (this corresponds to Re λ 0). Due to our assumptions that ω m and ≤ ≤ → λ 2mi, one has → Re((ω iλ)2 m2) = 8m2 + O(Re λ)+ O(Im λ 2m)+ O(ǫ2). (5.24) − − − Therefore, by (5.22),

Re ζ = ǫ−1 Re (ω iλ)2 m2 = O(ǫ−1), (5.25) − − p 31 showing that for ǫ sufficiently small one has ζ C D (we take ǫ > 0 smaller if necessary). Since we only ∈ \ 1 2 consider z D , the relation (5.18) yields Re λ z ǫ2 ǫ2, and then (5.22) and (5.25) lead to ∈ 1 | | ≤ | | ≤ 1 Im((ω iλ)2 m2) O(Re λ) Im ζ = Im ζ2 = O(ǫ) − − = O(ǫ) = O(ǫ), 2 Re ζ ǫ2 ǫ2 showing that the condition (5.23) holds true for ǫ sufficiently small, satisfying assumptions of Proposition A.4. Then, by Proposition A.4, there is C > 0 such that for any Ψ L2,−κ(Rn, C2N ) the map (5.20) satisfies ∈

Φ(Ψ,ǫ,z)) 2,−κ Rn C2N Cǫ Ψ 2,−κ Rn C2N , ǫ (0, ǫ ), z D . (5.26) k kL ( , ) ≤ k kL ( , ) ∈ 2 ∈ 1 We take ǫ smaller if necessary so that ǫ 1/(2C) (with C > 0 from (5.26)); then the linear map 2 2 ≤ I Φ( ,ǫ,z) : Y Y Φ(Y,ǫ,z) − · 7→ − is invertible, with

−1 (I Φ( ,ǫ,z)) 2,−κ 2,−κ 2, ǫ (0, ǫ ), z D . (5.27) k − · kL →L ≤ ∈ 2 ∈ 1 Since Φ( ,ǫ,z) is linear, writing (5.21) in the form Y Φ(Y)=Φ(X), we can express Y = (I Φ)−1Φ(X). · − − Thus, for each ǫ (0, ǫ ) and z D , we may deﬁne the mapping (X,ǫ,z) Y, which we denote by ϑ: ∈ 2 ∈ 1 7→ ϑ( ,ǫ,z) : L2,−κ(Rn, R(π−)) L2,−κ(Rn, R(π+)), · → ϑ( ,ǫ,z) : X Y = (I Φ( ,ǫ,z))−1Φ(X,ǫ,z). (5.28) · 7→ − ·

By (5.26) and (5.27), one has ϑ( ,ǫ,z) 2,−κ 2,−κ 2Cǫ, for ǫ (0, ǫ ) and z D . k · kL →L ≤ ∈ 2 ∈ 1 Finally, let us discuss the differentiability of ϑ with respect to z. The map Φ can be differentiated in the strong sense with respect to z. First, we notice that, by (5.22) and then by (5.18),

~~(ω iλ)2 m2 2(ω iλ) 2 ζ∂ ζ = ∂ − − = − ∂ (2ω + ǫ2z) = 2 ω iλ , (5.29) | z | z ǫ2 ǫ2 z | − |~~

with the right-hand side bounded uniformly in ǫ (0, ǫ ) and z D . Therefore, using the bound for the ∈ 2 ∈ 1 derivative of the analytic continuation of the resolvent (cf. Proposition A.4, which we apply with ν = 0 and also with ν = 1 to accommodate the operator ǫD0 from the deﬁnition of Φ in (5.20)), we conclude that there is C > 0 such that

C ω iλ 3 ∂ Φ( ,ǫ,z) 2,−κ 2,−κ ∂ Φ 2,−κ 2,−κ ∂ ζ | − | Cǫ k z · kL →L ≤ k ζ kL →L | z |≤ ζ 2 ζ ≤ h i | | for all ǫ (0, ǫ ) and z D . Above, ∂ is considered as a gradient in R2 = C; we used (5.29) and the ∈ 2 ∈ 1 z ∼ estimate (5.25). Then it follows from (5.28) that there is C > 0 such that one also has

~~3 ∂ ϑ( ,ǫ,z) 2,−κ 2,−κ Cǫ , k z · kL →L ≤ for all ǫ (0, ǫ ) and z D . One can see from (5.20) that Φ is analytic in the complex parameter z, hence ∈ 2 ∈ 1 so is ϑ.~~

~~As long as j N is sufﬁciently large so that ǫ (0, ǫ ) and z D , then, by Lemma 5.4, the relations ∈ j ∈ 2 j ∈ 1 (3.19) and (3.20) allow us to express~~

~~+ − π Ψj = ϑ(π Ψj,ǫj, zj), with ϑ( ,ǫ,z) a linear map from L2,−κ(Rn, C2N ) into itself, with ·~~

~~ϑ( ,ǫ,z) 2,−κ Rn C2N 2,−κ Rn C2N Cǫ. k · kL ( , )→L ( , ) ≤ 32 Now (3.17) and (3.18) can be written as~~

ǫ D π−Ψ + (m ω iλ )π−Ψ + ǫ2π−Vϑπ−Ψ = 0, (5.30) j 0 A j − j − j P j j P j ǫ D π−Ψ (m + ω + iλ )π−Ψ + ǫ2π−Vϑπ−Ψ = 0, (5.31) j 0 P j − j j A j j P j where Vϑ = Vϑ(y,ǫ , λ ), with Vϑ(y,ǫ,z) := V(y,ǫ) (1 + ϑ( ,ǫ,z)) satisfying j j ◦ · ϑ V (ǫ, λ) 2 2 V(ǫ) 2,−κ 2 1+ ϑ( ,ǫ,z) 2,−κ 2,−κ C, k kL →L ≤ k kL →L k · kL →L ≤ so that

~~− + − − ϑ − V(ǫj)Ψj = V(ǫj)(π Ψj + π Ψj)= V(ǫj) π Ψj + ϑ(π Ψj,ǫj, zj) = V π Ψj.~~

We recall the deﬁnition Z = (2ω + iλ )/ǫ2 (cf. (5.1)) and rewrite (5.30), (5.31), as the following system: j − j j j − m+ωj Vϑ − − −1D Vϑ − − πP ( 2 + Zj + )πP πP (ǫj 0 + )πA π Ψj ǫj P = 0, (5.32) − −1 ϑ − − 1 ϑ − −Ψ " π (ǫ D0 + V )π π (Zj + V )π # πA j A j P A − m+ωj A with j N. We rewrite the above as the nonlinear eigenvalue problem ∈ π−Ψ A(ǫ, z) P = 0, π−Ψ A π−( m+ω + z + Vϑ)π− π−(ǫ−1D + Vϑ)π− A(ǫ, z) := P ǫ2 P P 0 A , (5.33) π−(ǫ−1D + Vϑ)π− π−(z 1 + Vϑ)π− A 0 P A − m+ω A where the operator A(ǫ, z) : H1(Rn, R(π−)) L2(Rn, R(π−)) is considered as → A(ǫ, z) : L2 Rn, R(π−) R(π−) L2 Rn, R(π−) R(π−) , P × A → P × A with domain D(A(ǫ, z)) = H1(Rn, R(π−) R(π−)). Note that the operator A(ǫ, z)depends on z analyti- P × A cally via ϑ (cf. Lemma 5.4). By Weyl’s theorem,

σ (A(ǫ, z)) = , (m + ω)−1 + z (m ω)−1 + z, + , (5.34) ess −∞ − ∪ − ∞ so that 0 σ (A(ǫ, z)). Thus, the values Z deﬁned in (5.1) are such that the kernel of A(ǫ , z) is nontrivial 6∈ ess j j at z = Zj; such values of z are called the characteristic roots (or, informally, nonlinear eigenvalues) of A(ǫ, z). Nonlinear eigenvalue problem. We will study the location of characteristic roots of A(ǫ, z) using the theory developed by M. Keldysh [Kel51, Kel71]; see also [MS70, GS71]. Let X be a Hilbert space and Ω C ⊂ be an open neighborhood. We assume that the family of closed operators A(z) : X X, z Ω, is → ∈ a holomorphic family of type (A) [Kat76, Section VII-2]: that is, we assume that operators have the same domain D(A(z)) = D, z Ω, with D dense in X, and that for any u D the vector-function A(z)u is ∀ ∈ ∈ holomorphic in Ω. It follows that for any u, v D, each z Ω, and each η in the resolvent set of A(z ), ∈ 0 ∈ 0 the function u, (A(z) η)−1v is analytic in z in an open neighborhood of z and that A(z) is resolvent- h − i 0 continuous in z (in the norm topology).

Deﬁnition 5.6. The point z Ω is said to be regular for the operator-valued analytic function A(z) if the 0 ∈ operator A(z ) has a bounded inverse. If the equation A(z )ϕ = 0 has a non-trivial solution ϕ D, then z 0 0 0 ∈ 0 is said to be a characteristic root of A and ϕ0 an eigenvector of A corresponding to z0. Assumption 5.7. z Ω is normal characteristic root of A(z). 0 ∈

33 This means that A(z) has a bounded inverse for 0 < z z < R, with some R> 0, and 0 σ (A(z )); | − 0| ∈ d 0 see [Kel51, Kel71]. Due to Assumption 5.7, there is δ > 0 such that D σ(A(z )) = 0 . Due to the resolvent continuity of δ ∩ 0 { } A in z, there is an open neighborhood U Ω of z such that ∂D ρ(A(z)) for all z U. Let ⊂ 0 δ ⊂ ∈ 1 P = (A(z) η)−1 dη, z U. δ,z −2πi − ∈ I|η|=δ One has rank P = rank P < (the last inequality due to the assumption that 0 σ (A(z ))). δ,z δ,z0 ∞ ∈ d 0 Deﬁnition 5.8. The multiplicity α N of the characteristic root z of A(z) is the order of vanishing of ∈ 0 det A(z) R at z0. | (Pδ,z ) Remark 5.9. The above deﬁnition does not depend on a particular choice of δ > 0.

Lemma 5.10. Let z be a normal characteristic root of A(z) of multiplicity α N. The geometric multiplicity 0 ∈ of 0 σ (A(z )) satisﬁes g α. ∈ d 0 ≤ Proof. We denote ν = dim R(P ) < and choose the basis ψ in R(P ) so that ψ , 1 i g, δ,z0 ∞ { i}1≤i≤ν δ,z0 i ≤ ≤ are eigenvectors corresponding to zero eigenvalue of A(z0). Let A(z) be the matrix representation of A(z) in the basis Pδ,zψi 1≤i≤ν . Then the ﬁrst g columns of A(z) vanish at z = z0, hence det A(z) = O((z g { } − z0) ). The sum of multiplicities of characteristic roots is stable under perturbations (cf. [GS71, Theorem 2.1]):

Theorem 5.1. Let A(ǫ, z) : X X, ǫ R, z Ω, be a continuous family of operators with domains → ∈ ∈ D(A(ǫ, z)) = D, (ǫ, z) R Ω, which is resolvent-continuous in ǫ and z and analytic in z Ω. Let z ∀ ∈ × ∈ 0 be a normal characteristic root of A(0, z) of multiplicity α N. There is an open neighborhood Ω C of ∈ ⊂ z such that if ǫ 0 is sufﬁciently small, then the sum of multiplicities of all characteristic values of A(ǫ, z) 0 ≥ inside Ω equals α.

Proof. Fix δ > 0 sufﬁciently small so that σ(A(0, z )) D = 0 . Due to the resolvent continuity, there 0 ∩ δ { } is ǫ > 0 and an open neighborhood Ω C of z such that for all ǫ [0,ǫ ] and all z Ω one has 1 ⊂ 0 ∈ 1 ∈ σ(A(ǫ, z)) D = 0 . Let Γ= ∂D and denote ∩ δ { } δ 1 P (ǫ, z)= (A(ǫ, z) ηI)−1 dη, ǫ [0,ǫ ], z Ω. Γ −2πi − ∈ 1 ∈ IΓ Due to the resolvent continuity of A(ǫ, z) in ǫ and z, one has

~~rank P (ǫ, z) = rank P (0, z ) =: ν, ǫ [0, 1], z Ω. Γ Γ 0 ∀ ∈ ∀ ∈ R Let ψi 1≤i≤ν be the basis in (PΓ(0, z0)). Denote~~

ψ (ǫ, z)= P (ǫ, z)ψ , 1 i ν, ǫ [0, 1], z Ω; i Γ i ≤ ≤ ∈ ∈ it is a basis in R(PΓ(ǫ, z)) (we take Ω and ǫ1 > 0 smaller if necessary). Let M(ǫ, z) be the matrix representation of A(ǫ, z), restricted onto R(P (ǫ, z)), in this basis. If ǫ [0,ǫ ] and z Ω is a characteristic root of Γ ∈ 1 j ∈ A(ǫ, z), then its multiplicity equals the order of vanishing of det M(ǫ, z) at zj. Now the proof follows from Cauchy’s argument principle applied to det M(ǫ, z) in Ω when we take Ω and ǫ1 > 0 small enough so that zeros of det M(ǫ, z) do not intersect ∂Ω.

34 Now we apply the above theory to the operator A(ǫ, z) defined in (5.33); first, we will do the reduction of A using the Schur complement of its invertible block. By (5.34), we may assume that there is a sufficiently small open neighborhood U of z = 0 and that ω (0,m) is sufficiently large so that ∗ ∈ σ (A(ǫ, z)) D = ω (ω ,m), ǫ (0,ǫ ), z U, ess ∩ 1/(4m) ∅ ∀ ∈ ∗ ∀ ∈ ∗ ∀ ∈ where ǫ = m2 ω2. Since 0 σ (l ), with l from (2.18), we may assume that the open neighborhood ∗ − ∗ ∈ d − − U 0 is small enough so that ∋ { } p U σ(l )= 0 . (5.35) ∩ − { } Let Aij(ǫ, z), i, j = 1, 2, denote the operators which are the entries of A(ǫ, z) in (5.33), so that A(ǫ, z) = A (ǫ, z) A (ǫ, z) 11 12 . We then have: A (ǫ, z) A (ǫ, z) 21 22 −1 A12 H1→L2 + A21 H1→L2 + A21 L2→H−1 = O(ǫ ), k k k k k k (5.36) −2 A 2 2 = O(ǫ ), k 11kL →L and, taking ǫ∗ > 0 smaller if necessary, one has 2 −1 ǫ 4 A = + O 2 2 (ǫ ), ǫ (0,ǫ ), z U, (5.37) 11 2m L →L ∀ ∈ ∗ ∀ ∈ 4 with the estimate O 2 2 (ǫ ) uniform in z U. This suggests that we study the invertibility of A(ǫ, z) in L →L ∈ terms of the Schur complement of A11(ǫ, z), which is defined by S(ǫ, z)= A A A−1A : H1(Rn, R(π−)) H−1(Rn, R(π−)), (5.38) 22 − 21 11 12 A → A where A , A , A , and A are evaluated at (ǫ, z) (0,ǫ ) U. Let us derive the explicit expression for 11 12 21 22 ∈ ∗ × S(ǫ, z): 1 S(ǫ, z)= π− z + Vϑ π− A − m + ω A π−(D + ǫVϑ)π−m + ω + ǫ2(Vϑ + z) −1π−(D + ǫVϑ)π− − A 0 P P 0 A 1 ∆ = π− z + u2κ+ y π− + π− V ϑ u2κ π− A − m + ω κ m + ω A A − κ A D2 1 +π− 0 π− π−(D + ǫVϑ)π− π−(D + ǫVϑ)π− A m + ω A − A 0 P m + ω P 0 A 1 ǫ2(Vϑ + z) −1 π−(D + ǫVϑ)π− 1+ 1 π−(D + ǫVϑ)π−. − A 0 P m + ω m + ω − P 0 A Above, uκ = uκ(x) is the ground state of the nonlinear Schrödinger equation (2.2). We note that, by (3.13), − ϑ − − − − 2κ π V π = π V (1 + ϑ)π = π u + O 2 2 (ǫ); A A A ◦ A A κ L →L 1 ∞ 1+ we used the bounds πAφω L = O(ǫ κ ) (cf. Theorem 2.1) and ϑ 2 2 = O(ǫ) (cf. Lemma 5.4) k k k kL−s→L−s which yield V ϑ L2→L2 V 2 2 ϑ 2 2 = O(ǫ) for s > 1/2. Thus, taking into account k ◦ k ≤ k kL−s→Ls k kL−s→L−s Lemma B.4, the operator S(ǫ, z) defined in (5.38) takes the form

− 1 2κ ∆y − S(ǫ, z)= π z + u + + O 1 −1 (ǫ) π , ǫ [0,ǫ ), (5.39) A − 2m κ 2m H →H A ∈ ∗ with the estimate O 1 −1 (ǫ) uniform in z U. Above, we extended S(ǫ, z) in (5.39) from ǫ (0,ǫ ) to H →H ∈ ∈ ∗ ǫ [0,ǫ ) by continuity. ∈ ∗ The following lemma allows us to reduce the problem of studying the characteristic roots of A(ǫ, z) (see (5.33)) to the characteristic roots of S(ǫ, z) (cf. (5.38)).

35 Lemma 5.11. If ǫ > 0 is sufﬁciently small, then for all ǫ [0,ǫ ) the point z D is a characteristic ∗ ∈ ∗ 0 ∈ 1/(2m) root of A(ǫ, z) if and only if it is a characteristic root of S(ǫ, z).

Proof. Since the operator A (ǫ, z) : L2(Rn, R(π−)) L2(Rn, R(π−)) is invertible for ǫ > 0 small 11 P → P enough (see (5.37)), the expression for the Schur complement (D.4) and the relation (D.2) shows that the operator A(ǫ, z) from (5.33) is invertible if and only if so is S(ǫ, z).

Lemma 5.12. The multiplicity of the characteristic root z0 = 0 of S(0, z) is α = N/2. Proof. By (5.39), S(0, 0) = (z l )π−. Since dim ker(l ) = 1, one has dim ker(S(0, 0)) = rank π− = − − A − A N/2. Let e C2N , 1 i N/2, be the basis in R(π−); then u e is the basis in ker(S(0, 0)). i ∈ ≤ ≤ A κ i 1≤i≤N/2 We do not need to use the Riesz projectors since the operator is invariant in this space, being repre- S(0, z) S S N/2 sented by (0, z)= zIN/2; thus, det (0, z)= z . Lemma 5.13. There is no sequence of characteristic roots Z = 0 of A(ǫ , z) such that Z 0 as j . j 6 j j → →∞ Proof. Let δ > 0 be such that ∂D ρ(A(ǫ, z )), z = 0. Due to the continuity of the resolvent in z and ǫ, δ ⊂ 0 0 there is ǫ (0, ǫ ) and an open neighborhood U D , z U, such that ∂D ρ(A(ǫ, z)) for all ∗ ∈ 2 ⊂ 1/(2m) 0 ∈ δ ⊂ z U and ǫ [0,ǫ ). ∈ ∈ ∗ By (5.35) and Lemma 5.12, the sum of multiplicities of the characteristic roots of S(0, z) in U equals N/2, and by Theorem 5.1 the same is true for S(ǫ, z) for all ǫ [0,ǫ ). At the same time, by Lemma 3.4 and ∈ ∗ Lemma 5.10, z = 0 is a characteristic root of S(ǫ, z), ǫ [0,ǫ ), of multiplicity at least α = N/2. Hence, ∈ ∗ there can be no other, nonzero characteristic roots z U of S(ǫ, z) for any ǫ [0,ǫ ), and, in particular, ∈ ∈ ∗ given a sequence ǫ 0, there is no sequence of characteristic roots Z of S(ǫ , z) such that Z = 0 for j → j j j 6 j N, Z 0 as j . By Lemma 5.11, the same conclusion holds true for A(ǫ, z). ∈ j → →∞ By Lemma 5.13, Z = 0 for all but ﬁnitely many j N. By(2.19), this implies that λ = 2ω i for all but j ∈ j j ﬁnitely many j N. This concludes the proof of Theorem 2.3 (3). ∈ The proof of Theorem 2.3 is now complete.

~~36 A Appendix: Analytic continuation of the free resolvent~~

Let us remind the limiting absorption principle for the free resolvent [Agm75, Remark 2 in Appendix A] and [JK79, Theorem 8.1]. Lemma A.1 (Limiting absorption principle for the Laplace operator). Let n 1. For any k N , ν 2+2k, ≥ ∈ 0 ≤ s> 1/2+ k, and δ > 0, there is C = C(n,s,k,ν,δ) < such that ∞ k −1 −(k+1−ν)/2 ∂ ( ∆ z) 2 Rn ν Rn C z , z C (Dδ R+). k z − − kLs( )→H−s( ) ≤ | | ∈ \ ∪ Proof. For ν = 0, the lemma rephrases [JK79, Theorem 8.1] (stated for n = 3) or [Agm75, Theorem A.1 and Remark 2 in Appendix A]. The recurrence based on the identities

~~∆( ∆ z)−1 =1+ z( ∆ z)−1 and ∂k( ∆ z)−1 = k!( ∆ z)−k−1, k N , − − − − − z − − − − ∈ 0 provides all other cases.~~

Let us remind the limiting absorption principle for the resolvent of the free Laplacian. Lemma A.2 (Limiting absorption principle for the Laplace operator). Let n 1. For any s > 1/2, δ > 0, ≥ and ν N , ν 2, there is C = C(n,s,δ,ν) < such that ∈ 0 ≤ ∞ −1 −(1−ν)/2 ( ∆ z) 2 Rn ν Rn C z , z C (Dδ R+). k − − kLs( )→H−s( ) ≤ | | ∀ ∈ \ ∪ For the proof, see [Agm75, Theorem A.1 and Remark 2 in Appendix A]. We start with the following result: Lemma A.3. Let n 1. For any s > 1/2, δ > 0, and ν N , ν 2, there is C = C(n,s,δ,ν) > 0 such ≥ ∈ 0 ≤ that for any u H2(Rn) one has ∈ −(1−ν)/2 u ν C z ( ∆ z)u 2 , z C D . k kH−s ≤ | | k − − kLs ∀ ∈ \ δ This lemma follows from [Agm75, Theorem A.1 and Remark 2 in Appendix A]. It also implies the limiting absorption principle for the resolvent of the free Laplace operator (see e.g. [JK79, Theorem 8.1]), showing that the mapping

R0(z) = ( ∆ z)−1 : L2(Rn) L2 (Rn), s> 1/2, z C R , − − s → −s ∈ \ + is bounded uniformly in z C R , z δ, for each δ > 0, with the norm improving as z . ∈ \ + | |≥ | |→∞ Let E : L2(Rn) L2(Rn) denote the operator of multiplication by e−µhri, µ R. Following [Rau78], µ → ∈ not to confuse the regularized resolvent

R0 (ζ2) := E R0(ζ2)E = E ( ∆ ζ2)−1E , ζ C, Im ζ > 0 µ µ µ µ − − µ ∈ 0 with its analytic continuation through the line Im ζ = 0, we will denote the latter by Fµ (ζ). Proposition A.4 (Analytic continuation of the resolvent). Let n 1, µ> 0. ≥ 0 1. There is an analytic function Fµ (ζ),

F 0 : Im ζ > µ ( iR ) B(L2(Rn),L2(Rn)), µ { − } \ − + −→ such that F 0(ζ) = R0 (ζ2) for Im ζ > 0, and for any δ > 0 and ν N , ν 2, there is C = µ µ ∈ 0 ≤ C(n,µ,δ,ν) < such that ∞ k 0 C ∂ F (ζ) 2 ν , Im ζ µ + δ, dist(ζ, iR ) > δ. (A.1) k ζ µ kL →H ≤ (1 + ζ )k+1−ν ≥− − + | | 37 2. If n is odd and satisﬁes n 3, then (A.1) holds for all ζ C, Im ζ µ + δ. ≥ ∈ ≥− Remark A.5. This result in dimension n = 3 was stated and proved in [Rau78, Proposition 3], as a conse- 0 2 quence of the explicit expression for the integral kernel of Rµ(ζ ),

e−µhyieiζ|y−x|e−µhxi , Im ζ > 0, x,y R3, − 4π y x ∈ | − | which could be extended analytically to the region Im ζ > µ as a holomorphic function of ζ with values − in L2(R3 R3). In [Rau78], the restriction on ζ was stronger: Im ζ > µ/2+ δ (with any δ > 0); this × − was a pay-off for using an elegant argument based on the Huygens principle. (We note that our signs and inequalities are often the opposite to those of [Rau78] since we consider the resolvent of ∆ instead of ∆.) − 0 2 Rn 2,µ Rn Proof. Let us deﬁne the analytic continuation of Fµ (ζ). For u, v L ( ) we deﬁne uµ, vµ L ( ) by −µhxi −µhxi ∈ ∈ uµ(x)= e u(x), vµ(x)= e v(x) and consider 1 dnξ 0 (A.2) I(ζ)= v, Fµ (ζ)u = vµ(ξ) 2 2 uµ(ξ) n , h i Rn ξ ζ (2π) Z − which is an analytic function in ζ C+ := z C :b Im z > 0 . c ∈ { ∈ } Let us prove analyticity in ζ for Im ζ > µ, Re ζ > 0 (the case Re ζ < 0 is considered similarly). It is − enough to prove that for any a > 0 and any δ > 0, δ a/3, I(ζ) extends analytically into the rectangular neighborhood ≤

δ = ζ C ; a δ Re ζ a + δ, µ + δ Im ζ 0 (A.3) Ka { ∈ − ≤ ≤ − ≤ ≤ } (see Figure 2), satisfying there the bounds (A.1) with constants cj independent of a. ✻Im λ qa− qδ aq a+ qδ q ✲Re λ 0 a−2δ a+2δ δ Ka γa −µ+δ

−µ

~~δ Figure 2: The rectangular semi-neighborhood Ka around a surrounded by the contour γa = δ {λ : Im λ = ga(Re λ), a − 2δ ≤ Re λ ≤ a + 2δ}; dist(Ka,γa) ≥ δ/2.~~

We pick a> 0 and δ > 0, with a 3δ, and break the integral (A.2) into two: ≥ (δ) (δ) I(ζ)= I1 (ζ)+ I2 (ζ)= + . (A.4) Z||ξ|−a|>2δ Z||ξ|−a|<2δ The ﬁrst integral in (A.4) is ﬁnite, being bounded by 1 1 1 dnξ I(δ)(ζ) vˆ (ξ) uˆ (ξ) | 1 |≤ | µ || µ |2 ζ ξ ζ − ξ + ζ (2π)n Z ||ξ|−a|>2δ | | | |− | |

vˆ (ξ) uˆ (ξ) 2 dnξ u v | µ || µ | k µkk µk, ≤ 2 ζ δ (2π)n ≤ ζ δ RZn | | | | and therefore is analytic in ζ and is bounded by C/ ζ . Above, to estimate the denominators, we took into | | account that for ζ δ and ξ a > 2δ, ∈ Ka || |− | ξ ζ ( ξ a) + (a Re ζ) ξ a a Re ζ > 2δ δ = δ. || |± | ≥ | | |− ± | ≥ || |− | − | ± | − 38 To analyze the second integral in (A.4), we will deform the contour of integration in ξ. Let g C∞ (R) be 0 ∈ comp even, g 0, supp g [ 2δ, 2δ], with g (0) = µ + δ/2 and monotonically increasing to zero as t 2δ. 0 ≤ 0 ∈ − 0 − | |→ Moreover, we may assume that g′ < 4µ/δ and that dist(γ , δ) δ/2, where δ is deﬁned in (A.3) and | 0| 0 K0 ≥ Ka γ = (λ, g (λ)) : λ 2δ ; see Figure 2. Deﬁne g (t)= g (t a). 0 { 0 | |≤ } a 0 − 2,µ n µhri Lemma A.6. Let u L (R ), so that u 2,µ Rn := e u 2 Rn < . Then its Fourier transform, ∈ k kL ( ) k kL ( ) ∞ uˆ(ξ), can be extended analytically into the µ-neighborhood of Rn Cn, which we denote by ⊂ Ω (Rn)= ξ Cn ; Im ξ <µ Cn, µ { ∈ | | }⊂ and there is Cµ > 0 such that

~~uˆ 2 Rn C u 2,µ Rn , (A.5) k kL (Ωµ( )) ≤ µk kL ( ) Rn R2n Cn where Ωµ( ) is interpreted as a region in ∼= .~~

~~Proof. To prove this lemma, one deﬁnes uˆ on Ωµ as the Fourier–Laplace transform of u and then computes the L2-norm for ﬁxed Im ζ.~~

By Lemma A.6, the functions U(ξ)= u (ξ) and V (ξ)= v (ξ) could be extended analytically in ξ Rn µ µ ∈ into the strip ξ Cn, Im ξ < µ. We rewrite the second integral in (A.4) in polar coordinates, denoting ∈ | | λ = ξ [a 2δ, a + 2δ], θ = ξ/ ξ Sn−1, and then deform the contour of integration in λ, arriving at | |∈ − | |∈ c b V (θλ)U(θλ) dΩθ (δ) n−1 (A.6) I2 (ζ)= 2 2 λ dλ n , Sn−1 λ ζ (2π) Zγa× − with γ as on Figure 2. Clearly, (A.6) is analytic for Re ζ > 0 and Im ζ > 0 (since Im λ2 0 while a ≤ Im ζ2 > 0). Let us argue that (A.6) can also be extended analytically into the box δ . For λ γ and ζ δ , taking Ka ∈ a ∈ Ka into account that

λ ζ δ/2, λ + ζ Re λ + Re ζ (a 2δ) + (a δ) = 2a 3δ a | − |≥ | |≥ ≥ − − − ≥ (recall that δ a/3), we see that (A.6) deﬁnes an analytic function which is bounded by ≤ (δ) I2 (ζ) (A.7) | | 1 2 2 dΩθ dΩθ V (θλ) 2 λ n−1 dλ U(θλ) 2 λ n−1 dλ , ≤ aδ  | | | | | |(2π)n | | | | | |(2π)n  γ ×ZSn−1 γ ×ZSn−1  a a    where the integration in λ over the contour γa could be parametrized by

λ(t)= t + ig (t), t (a 2δ, a + 2δ), dλ = (1 + ig′ (t)) dt, (A.8) a ∈ − a so that dλ is understood as (1 + (g′ (t))2)1/2 dt. Our assumption that a 3δ allows us to bound the first | | a ≥ factor in (A.7) by 2 2 . Moreover, if ζ 2(µ + δ), the first factor in (A.7) is also bounded by aδ ≤ 3δ2 | |≥ 2 2 2 4 < < < , ζ δ D . aδ ( Re ζ δ)δ ( ζ µ δ)δ ζ δ ∀ ∈ Ka \ 2(µ+δ) | |− | |− − | | Therefore, that factor is bounded by c/(1 + ζ ) with certain c = c(µ, δ) > 0. To study the integrals in (A.7), | | we define G : Rn Rn by → η G(η)= g ( η )ρ( η /δ), η Rn, (A.9) η a | | | | ∈ | | 39 where ρ C∞(R) satisfies ρ(t) 1 for t 1, ρ(t) 0 for t 1/2, and parametrize ξ as follows: ∈ ≡ | |≥ ≡ | |≤ ξ = η + iG(η) Cn, η Rn; η a 2δ. ∈ ∈ || |− |≤

We have: n dΩθ 4µ 2 2 U(θλ) 2 λ n−1 dλ 1+ U(η + iG(η)) 2 dnη, | | | | | |(2π)n ≤ δ | | n−1 γa×ZS ||η|−Za|<2δ where we took into account that both λ/ Re λ and dλ/ Re dλ (which are understood with the aid of the 2 1/2 | | ′| 2 1/2 | parametrization (A.8) as (t + g(t) ) /t and (1 + (ga(t)) ) , respectively) are bounded by

1 + (g′ )2 1 + (4µ/δ)2. 0 ≤ q One has: p U(η + iG(η)) = Agu(η), where −ix·η x·G(η) −µhxi n Agu(η) := e e e u(x) d x Rn Z is an oscillatory integral operator with the non-degenerate phase function φ(x, η) = x η and a bounded G · smooth symbol a(x, η)= ex· (η)−µhxi. Note that, by (A.9), x G(η) µ x x g ( η ) µ x < 0, (x, η) Rn Rn. · − h i ≤ | || a | | |− h i ∀ ∈ × ∗ By the van der Corput-type arguments applied to AgAg [Ste93, Chapter IX], one can show that Ag is continuous in L2(Rn). Then it follows that there is c = c(µ, δ) > 0 such that

There is a similar bound for V . Thus, there is C = C(µ, δ) > 0 such that I(δ)(ζ) C(µ,δ) v u , which is | 2 |≤ |ζ|δ k kk k the desired bound. The estimates on ∂jF 0(ζ), j N, are proved similarly, writing out the derivatives of (ξ2 ζ2)−1 and ζ µ ∈ − proceeding with the same decomposition as in (A.4); the only difference is the contribution from higher powers of ξ2 ζ2 in the denominator. − This settles the ﬁrst part of Proposition A.4. Before we prove the second part of Proposition A.4, we need the following technical lemma. Lemma A.7. Let ρ> 0 and let N N be odd and satisfy N 3. The analytic function ∈ ≥ ρ λN−1 dλ F (ζ)= , ζ C, Im ζ > 0, N,ρ λ2 ζ2 ∈ Z0 − extends analytically into an open disc Dρ. Moreover, one has ρN−2 ρ + ζ F (ζ) 2+ln N + π + ln | | , ζ D . (A.10) | N,ρ |≤ 2 ρ ζ ∈ ρ − | | 2 2 Proof. Using the identity λ = 1+ ζ (note that the denominator is nonzero since λ 0 and Im ζ > 0) λ2−ζ2 λ2−ζ2 ≥ and remembering that N is odd, we have: ρ λN−1 dλ ρ ζN−1 F (ζ)= = λN−3 + ζ2λN−5 + + ζN−3 + dλ N,ρ λ2 ζ2 · · · λ2 ζ2 Z0 − Z0 − ρN−2 ζ2ρN−4 ζN−2 ρ ζ = + + + ζN−3ρ + Ln − + πi . (A.11) N 2 N 4 · · · 2 ρ + ζ − − 40 Above, Ln denotes the analytic branch of the natural logarithm on C R speciﬁed by Ln(1) = 0; we also \ − took into account that, since Im ζ > 0, λ ζ 2λ lim Ln − = lim Ln 1+ = Ln( 1 0i) = πi. λ→0+ λ + ζ λ→0+ − ζ − − − Due to the assumption N 3, the right-hand side of (A.11) extends to an analytic function of ζ as long as ≥ ζ D . The bound (A.10) immediately follows from the inequalities ∈ ρ N−2 1 1 1 1 1 1 ζ < ρ, 1+ + + + 1+ 1+ ln(N 2) | | 3 5 · · · N 2 ≤ 2 j ≤ 2 − − Xj=2 (with the understanding that the summation gives no contribution when N = 3), and the bound

ρ ζ ρ + ζ Ln − + πi π + ln | | ρ + ζ ≤ ρ ζ − | | ρ− ζ valid for ζ Dρ C+ (when arg ( π, 0)). ∈ ∩ ρ+ζ ∈ − Remark A.8. Note that the conclusion of the lemma would not hold if N were even: in that case, one arrives at functions which have a branching point at ζ = 0; e.g.

ρ λ dλ 1 ρ2 = ln 1 , λ2 ζ2 2 − ζ2 Z0 − ρ λ3 dλ ρ ζ2λ ρ2 ζ2 ρ2 = λ + dλ = + ln 1 , λ2 ζ2 λ2 ζ2 2 2 − ζ2 Z0 − Z0 − which behave like ln ρ and ζ2 ln ρ when ζ ρ (hence have a branching point at ζ = 0). − ζ − ζ | |≪ Now let us prove the second part of Proposition A.4; from now on, we assume that n is odd and satisﬁes n 3. It is enough to prove that the function I(ζ) deﬁned in (A.2) is analytic inside the disc D C. ≥ µ ⊂ We pick ρ (0,µ) and break the integral (A.2) into two parts: ∈

V (ξ)U(ξ) n (ρ) (ρ) I(ζ)= 2 2 d ξ = I1 (ζ)+ I2 (ζ), (A.12) Rn ξ ζ Z − where V (ξ)U(ξ) I(ρ)(ζ) := dnξ, (A.13) 1 ξ2 ζ2 Z|ξ|≤ρ − V (ξ)U(ξ) I(ρ)(ζ) := dnξ. (A.14) 2 ξ2 ζ2 Z|ξ|>ρ − The function I(ρ)(ζ) in (A.14) is analytic in the disc ζ D , and moreover for any r (0, ρ) one has 2 ∈ ρ ∈

~~(ρ) V (ξ)U(ξ) n 1 sup I2 (ζ) sup 2 2 d ξ 2 2 v L2 u L2 . ζ∈Dr | |≤ ζ∈Dr |ξ|>ρ ξ ζ ≤ ρ r k k k k Z − −~~

~~(ρ) Let us consider I (ζ) deﬁned in (A.13). Since both V (ξ) and U(ξ) are analytic for ξ Cn, ξ < µ, we 1 ∈ | | have the power series expansions~~

α α α V (ξ)= Vαξ ,U(ξ)= Uαξ ,V (ξ)U(ξ)= Cαξ , α∈Nn α∈Nn α∈Nn X0 X0 X0 41 which are absolutely convergent for ξ <µ. Denote λ = ξ , θ = ξ/ ξ Sn−1. Then | | | | | |∈ ρ |α|+n−1 (ρ) V (ξ)U(ξ) n α λ dλ θ θ I1 (ζ)= 2 2 d ξ = Cα dΩ 2 2 , (A.15) ξ ζ n 0 λ ζ α∈N − |ξ|≤Z ρ − X0 SnZ 1 Z − where θα = θα1 ...θαn , θ = (θ ,...,θ ) Sn−1. We note that, by parity considerations, the terms 1 n 1 n ∈ corresponding to at least one αj being odd are equal to zero, hence the summation in the right-hand side is only over α (2N )n. We claim that the series (A.15) deﬁnes an analytic function in D , and moreover for ∈ 0 ρ each r (0, ρ) there is C > 0 such that ∈ (ρ) sup I1 (ζ) C v L2(Rn) u L2(Rn). ζ∈Dr | |≤ k k k k

~~We have:~~

(ρ) α I1 (ζ) = Cα θ dΩθF|α|+n,ρ(ζ) Sn−1 N n α∈X(2 0) Z α |α| F|α|+n,ρ(ζ) = Cα θ dΩθR , (A.16) Sn−1 |α| N n R α∈X(2 0) Z where R (ρ, µ) and the analytic function F (ζ) was deﬁned in Lemma A.7. By that lemma, ∈ N,ρ n+|α|−2 ( α + n)F|α|+n,ρ(ζ) ρ ρ + ζ | | ( α + n) 2 + ln( α + n)+ π + ln | | R|α| ≤ | | 2R|α| | | ρ ζ − | | are analytic functions of ζ D , r (0, ρ), which are bounded uniformly in α Nn and ζ D , by some ∈ r ∈ ∈ 0 ∈ r cr,ρ,R > 0, 0

(ρ) α |α| D I1 (ζ) cr,ρ,R Cαθ R dΩθ, ζ r. (A.17) Sn−1 | |≤ N n | | ∈ α∈X(2 0) Z Now we can argue that the series (A.16) is absolutely convergent. To bound the right-hand side in (A.17), we use the following lemma with ξ = Rθ. α Lemma A.9. For any 0 0 such that for any analytic function a(ξ)= Nn aαξ , α∈ 0 ξ Dn Cn, which has ﬁnite norm in L1(B2n), where B2n R2n is identiﬁed with Dn Cn, one has ∈ µ ⊂ µ µ ⊂ µ ⊂ P α sup aαξ CR,µ a L1(B2n). Dn | |≤ k k µ ξ∈ R α∈Nn X0 Proof. One can prove Lemma A.9 using the Cauchy estimates

iϕ1 iϕn a(Me ,...,Me ) dϕ1 dϕn aα | | . . . , | |≤ M |α| 2π 2π S1 Z S1 M ×···× M with M (R,µ), and changing each of the integrations from ϕ over S1 to the integration over a thin closed ∈ j M annulus included in the region R< z <µ. | j|

~~42 This lemma, together with the estimate (A.5) from Lemma A.6, shows that, for ζ D , (A.17) is bounded ∈ r by~~

(ρ) Sn−1 α |α| I1 (ζ) cr,ρ,Rvol ( ) sup Cαθ R θ Sn−1 | | ≤ ∈ N n | | α∈X(2 0) n−1 cr,ρ,RCR,µvol (S ) VU 1 B2n ≤ k kL ( µ ) n−1 c C vol (S ) V 2 B2n U 2 B2n ≤ r,ρ,R R,µ k kL ( µ )k kL ( µ ) 2 n−1 c C C vol (S ) v 2,µ Rn u 2,µ Rn , ≤ r,ρ,R R,µ µ k kL ( )k kL ( ) where V (ξ) and U(ξ), ξ Ω (Rn) Cn, denote the analytic continuations of vˆ(ξ) and uˆ(ξ), ξ Rn, into ∈ µ ⊂ ∈ the µ-neighborhood of Rn in Cn. We conclude that the series (A.15) is absolutely convergent and therefore deﬁnes an analytic function. Thus, I(ρ)(ζ) (and hence I(ζ) in (A.12)) has an analytic continuation into the disc D for arbitrary ρ 1 ρ ∈ (0,µ), and for any r (0, ρ) the function I(ρ)(ζ) (and hence I(ζ)) is bounded by C(r) v u as long as ∈ 1 k kk k ζ D . This concludes the proof of Proposition A.4. ∈ r B Appendix: Spectrum of the linearized nonlinear Schrodinger¨ equation

For the nonlinear Schrödinger equation and several similar models, real eigenvalues could only emerge from the origin, and this emergence is controlled by the Kolokolov stability condition [Kol73]. Let us give the essence of the linear stability analysis on the example of the (generalized) nonlinear Schrödinger equation, 1 i∂ ψ = ∆ψ f( ψ 2)ψ, ψ(t,x) C, x Rn, n 1, t R, t −2m − | | ∈ ∈ ≥ ∈ where the nonlinearity satisfies f C∞(R), f(0) = 0. One can easily construct solitary wave solutions ∈ φ(x)e−iωt, for some ω R and φ H1(Rn): φ(x) satisfies the stationary equation ωφ = 1 ∆φ f(φ2)φ, ∈ ∈ − 2m − and can be chosen strictly positive, even, and monotonically decaying away from x = 0. The value of ω can not exceed 0; we will only consider the case ω < 0. We use the Ansatz ψ(t,x) = (φ(x)+ ρ(t,x))e−iωt, with ρ(t,x) C. The linearized equation on ρ is called the linearization at a solitary wave: ∈ 1 1 ∂ ρ = ∆ρ ωρ f(φ2)ρ 2f ′(φ2)φ2 Re ρ . (B.1) t i − 2m − − − Remark B.1. Because of the term with Re ρ, the operator in the right-hand side is R-linear but not C-linear. To study the spectrum of the operator in the right-hand side of (B.1), we first write it in the C-linear form, Re ρ(t,x) considering its action onto ρ(t,x)= : Im ρ(t,x) 0 l Re ρ(t,x) ∂ ρ = − ρ, ρ(t,x)= , t l 0 Im ρ(t,x) − + with 1 l = ∆ ω f(φ2), l = l 2φ2f ′(φ2). (B.2) − −2m − − + − − If φ S (Rn), then by Weyl’s theorem on the essential spectrum one has ∈ σ (l )= σ (l ) = [ ω , + ). ess − ess + | | ∞ 0 l Lemma B.2. σ − R iR. l+ 0 ⊂ ∪ − 43 2 0 l− l−l+ 0 Proof. We consider = . Since l− is positive-definite (φ ker(l−), being l+ 0 − 0 l+l− ∈ − nowhere zero, corresponds to the smallest eigenvalue), we can define the self-adjoint square root of l−; then

l 2 0 − l l l l l1/2l l1/2 R σd 0 = σd( − +) 0 = σd( + −) 0 = σd( − + − ) 0 , l+ 0 \{ } \{ } \{ } \{ }⊂ − 0 l with the inclusion due to l1/2l l1/2 being self-adjoint. Thus, any eigenvalue λ σ − satisfies − + − ∈ d l 0 − + λ2 R. ∈ Given the family of solitary waves, φ (x)e−iωt, ω R, we would like to know at which ω the ω ∈O⊂ eigenvalues of the linearized equation with Re λ > 0 appear. Since λ2 R, such eigenvalues can only be ∈ located on the real axis, having bifurcated from λ = 0. One can check that λ = 0 belongs to the discrete 0 l spectrum of − , with l 0 − + 0 l 0 0 l ∂ φ 0 − = 0, − − ω ω = , l 0 φ l 0 0 φ − + ω − + ω for all ω which correspond to solitary waves. Thus, if we will restrict our attention to functions which are 0 l spherically symmetric in x, the dimension of the generalized null space of − is at least two. Hence, l+ 0 the bifurcation follows the jump in the dimension of its generalized null space.− Such a jump happens at a 0 l ∂ φ particular value of ω if one can solve the equation − α = ω ω . This leads to the condition that l 0 0 − + ∂ φ 0 l φ ω ω is orthogonal to the null space of the adjoint to − , which contains the vector ω ; this 0 l 0 0 − + results in φ , ∂ φ = ∂ φ 2 /2 = 0. A slightly more careful analysis [CP03] based on construction h ω ω ωi ωk ωkL2 of the moving frame in the generalized eigenspace of λ = 0 shows that there are two real eigenvalues λ ± ∈ R that have emerged from λ = 0 when ω is such that ∂ φ 2 becomes positive, leading to a linear ωk ωkL2 instability of the corresponding solitary wave. The opposite condition, ∂ φ 2 < 0, is the Kolokolov ωk ωkL2 stability criterion which guarantees the absence of nonzero real eigenvalues for the nonlinear Schrödinger equation. It appeared in [Kol73, CL82, Sha83, Wei86, GSS87, BP92b] in relation to linear and orbital stability of solitary waves. For the applications to the nonrelativistic limit of the nonlinear Dirac equation, we need to consider the linearization of the nonlinear Schrödinger equation with pure power nonlinearity: f(τ)= τ κ, κ> 0: | | 1 iψ˙ = ∆ψ ψ 2κψ, ψ(t,x) C, x Rn. −2m − | | ∈ ∈ 0 l We need the detailed knowledge of the spectrum of the linearization operator − corresponding to l+ 0 −iωt − the linearization the solitary wave uκ(x)e , with uκ a strictly positive spherically symmetric solution to (2.2) and ω = 1 . − 2m Lemma B.3. The dimension of the null space and the generalized null space of the linearization operator 0 l − from (2.17) is given by l 0 − + l l ker 0 − L 0 − 2n + 2, κ = 2/n; l = n + 1, l = 6 + 0 + 0 (2n + 4, κ = 2/n. − − 44 Proof. Such computations have appeared in many articles. The relation (2.2) shows that l−uκ = 0. Taking i l the derivatives of this relation with respect to x , one also gets +∂xi uκ = 0, 1 i n. From [Kwo89] l 0 ≤ ≤ or [CGNT08, Lemma 2.1] we have that dim ker + = n + 1, hence there are no other vectors in 0 l − l 0 the kernel of + . 0 l − Now let us study the generalized eigenvectors. The relation l−uκ = 0 leads to

i 1 l (x u )= ∂ i u , 1 i n. − κ −m x κ ≤ ≤ This shows that l l 0 − ∂xi uκ 0 − 0 1 ∂xi uκ = 0, i = , 1 i n. (B.3) l+ 0 0 l+ 0 x uκ −m 0 ≤ ≤ − − We can not extend this sequence: there is no v such that

~~0 l v 0 − = , l 0 0 xiu − + κ i since x uκ is not orthogonal to the kernel of l+. Indeed, as follows from the identity~~

i i x u , ∂ i u = ( u x ∂ i u ), u (B.4) h κ x κi h − κ − x κ κi i 1 (no summation in i), one has x uκ, ∂xi uκ = 2 uκ, uκ < 0. h 1/κ i − h i By (2.2), the function uκ,λ(x)= λ uκ(λx) satisﬁes the identity

λ2 1 u = ∆u u1+2κ. −2m κ,λ −2m κ,λ − κ,λ Differentiating this identity with respect to λ at λ = 1 yields 1 1 1 0= l (∂ u )+ u = l u + x u + u . (B.5) + λ|λ=1 κ,λ m κ + κ κ ·∇ κ m κ This shows that 0 l 0 0 l θ 0 − = 0, − − = , (B.6) l 0 u l 0 0 u − + κ − + κ with m θ = u mx u , l θ = u . (B.7) − κ κ − ·∇ κ + κ 0 l The relations (B.3) and (B.6) show that dim L − 2n + 2. The dimension jumps above 2n + 2 l+ 0 ≥ in the case when one can ﬁnd α such that − 0 l 0 θ l α = θ, hence − = . (B.8) − l 0 α 0 − + This happens when θ in (B.7) is orthogonal to ker(l ) = Span u . Using the identity (B.4), we see that − { κ} 1 1 1 n θ, u = u x u , u = u , u + u , u . (B.9) mh κi − κ κ − ·∇ κ κ −κh κ κi 2 h κ κi D E The right-hand side of (B.9) vanishes when κ = 2/n (that is, when the nonlinear Schr¨odinger equation is charge-critical). We may choose α to be spherically symmetric so that it is orthogonal to ker(l+) =

45 m 2 Span ∂ i u ; 1 i n (as the matter of fact, one can take α(x)= x u (x)); then, by the Fredholm x κ ≤ ≤ − 2 κ alternative, there is β L2(Rn) such that ∈ 0 l β 0 l β = α, hence − − = . (B.10) + l 0 0 α − + (Let us also point out that α and β can be chosen real-valued.) This process can not be continued: there is no γ L2(Rn) such that l γ = β since β is never orthogonal to ker(l ); indeed, due to semi-positivity of l , ∈ − − − one has β, u = β, l θ = β, l l α = α, l α > 0. h κi h + i h + − i h − i We also need the following technical result. Lemma B.4. For z ρ(l ), the operator (l z)−1 : L2(Rn) H2(Rn) extends to a continuous mapping ∈ − − − → (l z)−1 : H−1(Rn) H1(Rn). − − → 2κ ∞ n Proof. Set a = sup Rn u (x) . Then there is C < such that for any ϕ C (R ) x∈ κ ∞ ∈ comp 2 1 1 1 2 ∞ n C ϕ 1 ϕ, (l + a)ϕ ϕ, ∆+ ϕ = ϕ 1 , ϕ C (R ), k kH ≥ |h − i| ≥ − 2m 2m 2mk kH ∀ ∈ comp D E hence the self-adjoint operator

l + a : H2(Rn) L2(Rn) (B.11) − → is positive-definite and invertible. We can extract its square root, which is a positive-definite bounded invertible operator (l + a)1/2 : H1(Rn) L2(Rn); − → then (B.11) also defines a bounded invertible operator (l + a)1/2 : H2(Rn) H1(Rn), and by duality − → there is also a bounded invertible mapping (l + a)1/2 : L2(Rn) H−1(Rn). We fix z ρ(l ); then the − → ∈ − mapping (l + a)1/2(l z)(l + a)−1/2 : H1(Rn) H−1(Rn) − − − − → is bounded and invertible. Since l + a and its square root commute with l z (when restricted e.g. to the − − − space of Schwartz functions), we apply the density argument to conclude that l z extends to a bounded − − invertible mapping H1(Rn) H−1(Rn). → C The limiting absorption principle for l − We will need the estimates on the resolvent of l = 1 ∆ u2κ from (2.18) in the spaces with exponential − 2m − 2m − κ weights. Lemma C.1. Let Ω C be a compact set. Assume that ⊂ Ω σ (l )= . ∩ d − ∅ If z = 1/(2m) is either an eigenvalue or a virtual level of l−, assume further that 1/(2m) Ω. 6∈ Then there is C = C(Ω) > 0 such that the map

uκ (l z)−1 uκ : L2(Rn) H2(Rn), z C σ(l ) κ ◦ − − ◦ κ → ∈ \ − satisfies κ −1 κ u (l z) u 2 2 C, z Ω [1/(2m), + ). k κ ◦ − − ◦ κkL →H ≤ ∀ ∈ \ ∞ 46 Remark C.2. The L2 L2 estimate on the resolvent improves as the spectral parameter goes to infinity, s → −s while L2 H2 estimate does not necessarily so (see e.g. Lemma A.3 with different values of ν); this is s → −s why in the above lemma we need to consider the spectral parameter restricted to a compact set Ω C. ⊂ Proof. The first part (bounds away from an open neighborhood of the threshold z = 1/(2m)) follows from [Agm75, Appendix A]. If the threshold z = 1/(2m) is a regular point of the essential spectrum of l− (neither an eigenvalue nor a virtual level), then the result follows from [JK79] for n = 3 and from [Yaf10, Proposition 7.4.6] for n 3. ≥ The case n 2, which is left to prove, follows from [JN01]. We recall the terminology from that article. ≤ Given the operators H = ∆ and H = H + V in L2(Rn), we denote 0 − 0 1,V 0; U = ≥ ; v = V 1/2, w = Uv, ( 1, V< 0 | | − M(ς)= U + v(H + ς2)−1v : L2(Rn) L2(Rn), Re ς > 0. 0 → There is the identity (1 w(H + ς2)−1v)(1 + w(H + ς2)−1v) = 1; hence, if M(ς) is invertible, − 0 1 w(H + ς2)−1v = (1+ w(H + ς2)−1v)−1 = (U + v(H + ς2)−1v)−1U = M(ς)−1U, − 0 0 U w(H + ς2)−1w = M(ς)−1, w(H + ς2)−1w = U M(ς)−1 − − 2κ κ (cf. [JN01, Equation (4.8)]). In the case at hand, V = uκ , U = 1, v = w = uκ (understood as operators of multiplication); thus, − − −

~~∆ −1 M(ς)= 1+ uκ + ς2 uκ, (C.1) − κ ◦ − 2m ◦ κ and, when M(ς) is invertible,~~

uκ (l z)−1 uκ = uκ ( ∆ u2κ + ς2)−1 uκ = 1 M(ς)−1, (C.2) κ ◦ − − ◦ κ κ ◦ − − κ ◦ κ − − with z and ς related by 1 z = ς2. 2m − Using the expression for the integral kernels of (H + ς2)−1 in dimensions n 2 (see [JN01, Section 3, 0 ≤ (3.13) and (3.14)]), the operator M(ς) extends by continuity to the region

ς C 0 ; Re ς 0 , { ∈ \ { } ≥ } with a singularity 1 log(ς) at ς = 0. If ∆ u2κ has no eigenvalue or virtual level at λ = 0, then − 2π − − κ M(ς) is invertible in the orthogonal complement of v (with the inverse bounded uniformly as ς 0), while → M(ς)/ log(ς) is always invertible (with the inverse bounded uniformly as ς 0) in the span of v. We deduce → that, as long as ς is small enough and ς = 0, M(ς) is invertible, with a uniform bound on M(ς)−1 in an | | 6 k k open neighborhood of 0 in the half-plane Re ς 0. Now the conclusion of the lemma for the case n = 2 follows from (C.2). The one-dimensional case is≥ dealt with similarly.

~~D The Schur complement~~

In the proofs, we often use the concept of the Schur complement, which is sometimes called the Feshbach map. Let us give the formulation in the operator form that we will need (see also [BFS98, Theorem IV.1] and [JN01, Lemma 2.3]).

47 Lemma D.1. Let X and X be Banach spaces and let A be a closed linear operator on X X with dense 1 2 1 ⊕ 2 domain D(A) X1 X2, such that ⊂ ⊕ A A A = 11 12 , A A 21 22 where A : X X , 1 i, j 2 are closed linear operators with domains ji i → j ≤ ≤ D(A )= x X ; (x , 0) D(A) , D(A )= x X ; (0,x ) D(A) j1 { 1 ∈ 1 1 ∈ } j2 { 2 ∈ 2 2 ∈ } which are dense in X1 and X2, respectively. Assume that there are Banach spaces W X Y , 1 i 2, i ⊂ i ⊂ i ≤ ≤ with continuous dense embeddings, such that A extends to a bounded operator from W to Y , with 1 ji i j ≤ i, j 2. Assume further that A : W Y has a bounded inverse. Let S be the Schur complement of ≤ 11 1 → 1 A11 deﬁned by S = A A A−1A : W Y . (D.1) 22 − 21 11 12 2 → 2 Then A (considered as an operator from D(A) to X1 X2) has a bounded inverse if and only if S : W2 ⊕ −1 −1 −1 → Y is invertible (not necessarily with the bounded inverse) and the restrictions S X , A A S X , 2 2 11 12 2 −1 −1 −1 −1 −1 | | S A21A X , and A A12S A21A X deﬁne the bounded operators 11 | 1 11 11 | 1 S−1 : X X , 2 → 2 A−1A S−1 : X X , 11 12 2 → 1 S−1A A−1 : X X , 21 11 1 → 2 A−1A S−1A A−1 : X X . 11 12 21 11 1 → 1 Proof. We consider A as a bounded operator from W W to Y Y . Since A : W Y has a 1 ⊕ 2 1 ⊕ 2 11 1 → 1 bounded inverse, the operators

I A−1A I 0 11 12 and 0 I A A−1 I 21 11 are bounded as linear operators in W W and in Y Y , respectively, with bounded inverses given by 1 ⊕ 2 1 ⊕ 2 I A−1A I 0 − 11 12 and . 0 I A A−1 I − 21 11 The following identities show that S is invertible as a map from W2 to Y2 if and only if A is invertible as a map from W W to Y Y : 1 ⊕ 2 1 ⊕ 2 A A I 0 A 0 I A−1A A = 11 12 = 11 11 12 , (D.2) A A A A−1 I 0 S 0 I 21 22 21 11 I 0 A A I A−1A A 0 11 12 − 11 12 = 11 . (D.3) A A−1 I A A 0 I 0 S − 21 11 21 22 Explicitly, the inverse of A in terms of the inverse of S is given by

−1 −1 −1 −1 −1 −1 A + A A12S A21A A A12S A−1 = 11 11 11 − 11 ; (D.4) −1 −1 −1 " S A21A S # − 11 this expression is known as the Banachiewicz inversion formula. One can see that A (considered as a map D X X −1 −1 −1 −1 −1 from (A) to 1 2) has a bounded inverse if and only if the mappings S , A11 A12S , S A21A11 , −1 −1 ⊕ −1 and A11 A12S A21A11 are bounded in appropriate spaces.

~~48 Remark D.2. Swapping the indices in Lemma D.1, we have the following result. Assume that A : W 22 2 → Y2 has a bounded inverse. Let T be the Schur complement of A22 deﬁned by~~

~~T = A A A−1A : W Y . (D.5) 11 − 12 22 21 1 → 1~~

Then A (considered as an operator from D(A) to X1 X2) has a bounded inverse if and only if T : W1 Y −1 −1 ⊕−1 −1 −1 −1 −1 −1 → 1 is invertible and the restrictions T X , A A21T X , T A12A X , and A A21T A12A X | 1 22 | 1 22 | 2 22 22 | 2 deﬁne the bounded operators

T −1 : X X , 1 → 1 A−1A T −1 : X X , 22 21 1 → 2 T −1A A−1 : X X , 12 22 2 → 1 A−1A T −1A A−1 : X X . 22 21 12 22 2 → 2 Similarly to (D.4), the inverse of A in terms of the inverse of T is given in the explicit form by

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