Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity

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Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity Nabile Boussaïd a , Andrew Comech b,c,∗ a b c

Université Bourgogne Franche-Comté, 25030 Besançon CEDEX, France Texas A&M University, College Station, TX 77843, USA Institute for Information Transmission Problems, Moscow 127051, Russia

a r t i c l e

i n f o

Article history: Received 23 March 2018 Accepted 23 July 2019 Available online xxxx Communicated by Fang Hua-Lin Keywords: Nonlinear Dirac equation Soler model Spectral stability Stability of solitary waves

a b s t r a c t We study the point spectrum of the linearization at a solitary wave solution φω (x)e−iωt to the nonlinear Dirac equation in Rn , for all n ≥ 1, with the nonlinear term given by f (ψ ∗ βψ)βψ (known as the Soler model). We focus on the spectral stability, that is, the absence of eigenvalues with positive real part, in the non-relativistic limit ω → m − 0, in the case when f ∈ C 1 (R \ {0}), f (τ ) = |τ |κ + O(|τ |K ) for τ → 0, with 0 < κ < K. For n ≥ 1, we prove the spectral stability of small amplitude solitary waves (ω  m) for the charge-subcritical cases κ  2/n (in particular, 1 < κ ≤ 2 when n = 1) and for the “charge-critical case” κ = 2/n (with K > 4/n). An important part of the stability analysis is the proof of the absence of bifurcation of nonzero-real-part eigenvalues from the embedded threshold points at ±2mi. Our approach is based on constructing a new family of exact bi-frequency solitary wave solutions in the Soler model and on the analysis of the behavior of “nonlinear eigenvalues” (characteristic roots of holomorphic operator-valued functions). © 2019 Elsevier Inc. All rights reserved.

* Corresponding author. E-mail address: [email protected] (A. Comech). https://doi.org/10.1016/j.jfa.2019.108289 0022-1236/© 2019 Elsevier Inc. All rights reserved.

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1. Introduction We study stability of solitary waves in the nonlinear Dirac equation with the scalar self-interaction [57,94], known as the Soler model: i∂t ψ = Dm ψ − f (ψ ∗ βψ)βψ,

ψ(t, x) ∈ CN ,

x ∈ Rn ,

n ≥ 1.

(1.1)

Here the Dirac operator is given by Dm = −iα·∇x +βm, with m > 0 and the self-adjoint 2 N × N Dirac matrices αi , 1 ≤ i ≤ n, and β, chosen so that Dm = −Δ + m2 ; for details, 1 see Notation at the end of this section. We assume that f ∈ C (R \ {0}) is real-valued, f (τ ) = |τ |κ + O(|τ |K ) as τ → 0, with 0 < κ < K. The structure of the nonlinearity, f (ψ ∗ βψ)βψ, is such that the equation is U(1)-invariant and hamiltonian. Given a solitary wave solution ψ(t, x) = φω (x)e−iωt to (1.1), with ω ∈ R and φω ∈ H 1 (Rn , CN ), we consider its perturbation, (φω (x) +ρ(t, x))e−iωt , and study the spectrum of the linearized equation on ρ (that is, the spectrum of the linearization operator). We will say that this particular solitary wave is spectrally stable if the spectrum of the linearization operator has no points in the right half-plane. In the present work, we prove the spectral stability of small amplitude solitary waves corresponding to the nonrelativistic limit ω  m,1 in the case κ  2/n, K > κ (“charge-subcritical”) and κ = 2/n, K > 4/n (“charge-critical”). This is the first 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 [27,82] 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 [97]). 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 [68] 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 [21, 55,89,104,105]. The nature of stability of solitary wave solutions of the nonlinear Dirac equation seems completely different from this picture [85, 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 [27,82] coming from higher-order integrals of motion. 1

We write ω  m to indicate that ω ∈ (m − ε, m) for some ε > 0 small enough.

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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 [6], 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 [74]. 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 [3,23,54]. In the absence of spectral stability, one expects to be able to prove orbital instability, in the sense of [55]; in the context of the nonlinear Schrödinger equation, such instability is proved in e.g. [51,61]. If instead a particular solitary wave is spectrally stable, one hopes to prove the asymptotic stability. We now 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 [91,92] for small amplitude solitary waves bifurcating from the ground state of the linear Schrödinger equation (thus with a potential) and [15–18] 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 [28,29,31]. 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 [19,72,87,93,98–101]. These have been improved in [30,32–36,48–50, 56,65–67,80,84,102]. This path is also developed for the nonlinear Dirac equation in [8, 9,13,26,37,83]. The needed dispersive properties of Dirac type models have been studied in these references and separately in [14,20,39–41,43–45,69–71,75]. 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. For the well-posedness problem, we also refer to the review [81]. The question of the existence of stationary solutions in the relation to the indefiniteness of the energy is discussed in the review [46]. While the purely imaginary essential spectrum of the linearization operator is readily available via Weyl’s theorem on the essential spectrum (see [10] 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 [10, Theorem 2.15], 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 ±(m + |ω|)i (see e.g. [3]), from the collisions of eigenvalues on the imaginary axis, from the embedded eigenvalues (by [10, 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 [24] and at λ = ±mi when ω = 0 [60].

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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 [24]; 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 [11]. By numerics of [38], in the case of the pure-power nonlinearity f (τ ) = |τ |κ with “super-critical” values κ > 2/n, the spectral instability disappears when ω ∈ (0, m) becomes sufficiently small (but may reappear for even smaller values of ω). 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 [52,79, 103,107,108]. On the contrary, for the nonlinear Dirac with the critical-power nonlinearity, the charge of solitary waves is no longer constant: by [11], one has ∂ω Q(φω ) < 0 for ω  m, where Q(φω ) is the corresponding charge (see (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. The bifurcations of eigenvalues from the origin are analyzed in Section 4. In Section 5, 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 [53,63,64,77] (this is done in Section 5.2). In Appendix A, we construct the analytic continuation of the resolvent of the free Laplace operator, extending the three-dimensional approach of [86] 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. Notation. We denote N = {1, 2, . . . } and N0 = {0} ∪ N. For ρ > 0, an open disc of radius ρ in the complex plane centered at z0 ∈ C is denoted by Dρ (z0 ) = {z ∈ C ; |z − z0 | < ρ}; we also write Dρ = Dρ (0). We denote r = |x| for x ∈ Rn , n ∈ N, and, abusing notations, we also denote the operator of multiplication with |x| and x = (1 +|x|2 )1/2 by r and r , respectively. For s, α ∈ R, we define the weighted Sobolev spaces of CN -valued functions by   Hsα (Rn , CN ) = u ∈ S  (Rn , CN ), uHsα < ∞ ,

uHsα =  r s −i∇ α uL2 .

We write L2s (Rn , CN ) for Hsα (Rn , CN ) with α = 0 and H α (Rn , CN ) for Hsα (Rn , CN ) with s = 0. For u ∈ L2 (Rn , CN ), we denote u = uL2 .

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For a pair of normed vector spaces E and F , B (E, F ) denotes the set of bounded linear operators from E to F . For an unbounded linear operator A acting in a Banach space X with dense domain D(A) ⊂ X, the spectrum σ(A) is the set of values λ ∈ C such that the operator A − λIX : 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) ; Aj v ∈ D(A) ∀j < k, Ak v = 0}. k∈N

k∈N

The discrete spectrum σd (A) is the set of isolated eigenvalues λ ∈ σ(A) of finite algebraic multiplicity: dim L(A − λ) < ∞. The essential spectrum σess (A) is the complementary set of the discrete spectrum in the spectrum. The point spectrum σp (A) is the set of eigenvalues (isolated or embedded into the essential spectrum). We denote the free Dirac operator by Dm = −iα · ∇x + βm = −i

n  i=1

αi

∂ + βm, ∂xi

m > 0,

and the massless Dirac operator by D0 = −iα·∇x . Above, αi and β are self-adjoint N ×N Dirac matrices which satisfy (αi )2 = β 2 = IN , αi αj + αj αi = 2δij IN , αi β + βαi = 0, 2 1 ≤ i, j ≤ n, so that Dm = (−Δ + m2 )IN . Here IN denotes the N × N identity matrix. The anticommutation relations lead to e.g. Tr αi = Tr β −1 αi β = − 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. By the Clifford algebra representation theory (see e.g. [47, Chapter 1, §5.3]), there is a relation N = 2[(n+1)/2] M , M ∈ N. Without loss of generality, we may   0 IN/2 . Then the anticommutation relations {αi , β} = 0, assume that β = 0 −IN/2   0 σi∗ i i 1 ≤ i ≤ n, show that the matrices α are block-antidiagonal: α = , 1 ≤ i ≤ n, σi 0 where the “generalized Pauli matrices” σi , 1 ≤ i ≤ n, of size (N/2) × (N/2) satisfy the relations σi∗ σj + σj∗ σi = 2δij , σi σj∗ + σj σi∗ = 2δij , 1 ≤ i, j ≤ n; the Dirac operator is thus given by    n   0 I 0 σi∗ , m > 0. Dm = −iα · ∇x + βm = −i ∂xi + m N/2 σi 0 0 −IN/2 i=1

(1.2) Acknowledgment. We are indebted to the anonymous referee for corrections and suggestions. The second author was partially supported by Université Bourgogne Franche– Comté and by the Metchnikov fellowship from the French Embassy. Both authors acknowledge the financial support of the région Bourgogne Franche–Comté.

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2. Main results We consider the nonlinear Dirac equation (1.1), i∂t ψ = Dm ψ − f (ψ ∗ βψ)βψ,

ψ(t, x) ∈ CN ,

x ∈ Rn ,

with the Dirac operator Dm of the form (1.2). Assumption 2.1. One has f ∈ C 1 (R \ {0}) ∩ C(R), and there are κ > 0 and K > κ such that |f (τ ) − |τ |κ | = O(|τ |K ),

|τ f  (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 bifurcations from the solitary wave solutions ϕω (y)e−iωt to the nonlinear Schrödinger equation 1 Δψ − |ψ|2κ ψ, iψ˙ = − 2m

ψ(t, y) ∈ C,

y ∈ Rn .

(2.1)

By [5,96] and [4] (for the two-dimensional case), the stationary nonlinear Schrödinger equation −

1 1 u=− Δu − |u|2κ u, 2m 2m

u(y) ∈ R,

y ∈ Rn ,

n≥1

(2.2)

has a strictly positive spherically symmetric exponentially decaying solution uκ ∈ C 2 (Rn ) ∩ H 1 (Rn )

(2.3)

(called the ground state) if and only if 0 < κ < 2/(n − 2) (any κ > 0 if n ≤ 2). Moreover, uκ ∈ H s (Rn ),

∀s < n/2 + 2

(2.4)

(see [11, Lemma A.1]) and there are constants 0 < cn,κ < Cn,κ < ∞ (which also depend on the value of the constant m > 0) such that cn,κ y −(n−1)/2 e−|y| ≤ uκ (y) ≤ Cn,κ y −(n−1)/2 e−|y| ,

∀y ∈ Rn ;

(2.5)

see e.g. [11, Lemma 4.5]. We set Vˆ (t) := uκ (|t|),

ˆ (t) := −(2m)−1 Vˆ  (t), U

t ∈ R;

(2.6)

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above, a spherically symmetric function uκ (y), y ∈ Rn , is understood as a function of ˆ ∈ C 1 (R) (which are even and odd, r = |y|. By (2.2), the functions Vˆ ∈ C 2 (R) and U respectively) satisfy 1 ˆ ˆ = |Vˆ |2κ Vˆ , ˆ + n − 1U V + ∂t U 2m t

ˆ = 0, ∂t Vˆ + 2mU

t ∈ R,

(2.7)

ˆ (t)/t at t = 0 is understood in the limit sense, limt→0 U ˆ (t)/t = U ˆ  (0). where U In the nonrelativistic limit ω  m, the solitary wave solutions to (1.1) are obtained ˆ ) [11]; we start with summarizing their asymptotics. as bifurcations from (Vˆ , U 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 ω1 ∈ (m/2, m) such that for all ω ∈ (ω1 , m) there are solitary wave solutions φω (x)e−iωt to (1.1), such that φω ∈ H 2 (Rn , CN ) ∩ C(Rn , CN ), ω ∈ (ω1 , m), with φω (x)∗ βφω (x) ≥ |φω (x)|2 /2

∀x ∈ Rn ,

∀ω ∈ (ω1 , m),

(2.8)

and  1  φω L∞ (Rn ,CN ) = O (m2 − ω 2 ) 2κ ,

ω  m.

More explicitly,  v(r, ω)ξ φω (x) = iu(r, ω) x · σ ξ , r 

x ∈ Rn ,

r = |x|,

ξ ∈ CN/2 ,

|ξ| = 1, (2.9) 1

where σ = (σ1 , . . . , σn ) are “generalized Pauli matrices” from (1.2), v(r, ω) =  κ V (r, ), √ 1 u(r, ω) = 1+ κ U (r, ), r ≥ 0, with  = m2 − ω 2 , limr→0 u(r, ) = 0, and V (t, ) = Vˆ (t) + V˜ (t, ),

ˆ (t) + U ˜ (t, ), U (t, ) = U

t ∈ R,

 > 0,

ˆ (t) defined in (2.6). with Vˆ (t), U There is b0 < ∞ such that |V (t, )| + |U (t, )| ≤ b0 t −(n−1)/2 e−|t| ,

∀t ∈ R,

∀ ∈ (0, 1 ).

(2.10)

  ˜ ˜ (t, ) = V (t, ) satisfies There are γ > 0 and b1 < ∞ such that W ˜ (t, ) U ˜ H 1 (R,R2 ) ≤ b1 2κ , eγt W with 1 =



∀ ∈ (0, 1 ),

(2.11)

m2 − ω21 and   κ := min 1, K/κ − 1 .

(2.12)

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There is a C 1 map ω → φω ∈ H 1 (Rn , CN ), with ˜ (·, ) ∈ H 1 (R) × H 1 (R) ∩ C 1 (R, R2 ), ∂ W even odd

∂ω φω ∈ H 1 (Rn , CN ),

1 1 where Heven (R) and Hodd (R) denote functions from H 1 (R) which are even and odd, respectively;

˜ (·, )H 1 (R,R2 ) = O(2κ−1 ), eγt ∂ W

 ∈ (0, 1 ),

(2.13)

and there is c > 0 such that ∂ω φω 2L2 (Rn ,CN ) = c−n+ κ (1 + O(2κ )), 2

ω=

m2 −  2 ,

 ∈ (0, 1 ).

Additionally, assume that κ, K from Assumption 2.1 satisfy either κ < 2/n, or κ = 2/n, K > 4/n. Then there is ωc < m such that ∂ω Q(φω ) < 0 for all ω ∈ (ωc , m). If instead κ > 2/n, then there is ωc < m such that ∂ω Q(φω ) > 0 for all ω ∈ (ωc , m). Above,

Q(ψ) =

ψ ∗ (t, x)ψ(t, x) dx

(2.14)

Rn

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 , |τ f  (τ ) − κ|τ |κ | ≤ C|τ |K ,

|f (τ )| ≤ (c + 1)|τ |κ ,

∀τ ∈ R,

(2.15)

|τ f  (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 ω1  m sufficiently close to m so that φω L∞ remains smaller than one for ω ∈ (ω1 , m). The eigenvalues of the linearization at solitary waves φω e−iωt with ω = ωj , ωj → m, can only accumulate to λ = ±2mi and λ = 0; see [10, 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 spherically 1 symmetric exponentially decaying solution to (2.2), then uκ (x)e−iωt with ω = − 2m is a

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solitary wave solution to the nonlinear Schrödinger equation (2.1). The linearization at this solitary wave (see (B.2)) is given by  ∂t

  Re ρ 0 = Im ρ −l+

l− 0



Re ρ Im ρ

 (2.17)

(see e.g. [68]), with l± defined by l− =

Δ 1 − − u2κ κ , 2m 2m

l+ =

Δ 1 − − (1 + 2κ)u2κ κ , 2m 2m

D(l± ) = H 2 (Rn ). (2.18)

Theorem 2.2 (Bifurcations from the origin at ω = m). Let n ≥ 1. Let f ∈ C 1 (R \ {0}) ∩ C(R) satisfy Assumption 2.1 with some values of κ, K. Let φω e−iωt , ω ∈ (ω1 , m), be the family of solitary wave solutions to (1.1) constructed in Theorem 2.1.   Let ωj j∈N , ωj ∈ (ω1 , m), be a sequence such that ωj → m, and assume that λj is an eigenvalue of the linearization of (1.1) at the solitary wave φωj (x)e−iωj t (see Section 3) and that Re λj = 0 for all j ∈ N, λj → 0. Denote Λj =

λj ∈ C, 2j

j = (m2 − ωj2 )1/2 ,

j ∈ N,

and let Λ0 ∈ C ∪ {∞} be an accumulation point of the sequence (Λj )j∈N . Then:  0 l− ∩ R. In particular, Λ0 = ∞. 1. Λ0 ∈ σp −l+ 0   2. Λ0 = 0 could be an accumulation point of Λj j∈N only when κ = 2/n and ∂ω Q(φω ) > 0 for ω ∈ (ω∗ , m), with some ω∗ < m. If, moreover, Λj → Λ0 = 0, then λj ∈ R for all but finitely many j ∈ N. 

In other words, as long as ∂ω Q(φω ) < 0 for ω  m, there can be no linear instability due to bifurcations from the origin: there would be no eigenvalues λj of the linearization at solitary waves with ωj → m such that Re λj = 0, λj → 0. We prove Theorem 2.2 in Section 4. Theorem 2.3 (Bifurcations from ±2mi at ω = m). Let n ≥ 1. Let f ∈ C 1 (R \ {0}) ∩ C(R) satisfy Assumption 2.1 with some values of κ, K. Let φω (x)e−iωt , ω ∈ (ω1 , m), be the family of solitary wave solutions to (1.1) constructed in Theorem 2.1.   Let ωj j∈N , ωj ∈ (ω1 , m), be a sequence such that ωj → m and assume that λj is an eigenvalues of the linearization of (1.1) at the solitary wave φωj (x)e−iωj t (see Section 3) and that λj → 2mi. Denote Zj = −

2ωj + iλj ∈ C, 2j

j = (m2 − ωj2 )1/2 ,

j ∈ N,

and let Z0 ∈ C ∪ {∞} be an accumulation point of the sequence (Zj )j∈N . Then:

(2.19)

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1 1. Z0 ∈ { 2m } ∪ σd (l− ). In particular, Z0 = ∞. 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 Z0 = 1/(2m). 3. If Zj → Z0 and Z0 = 0, then λj = 2ωj i for all but finitely many j ∈ N.

We note that, according to the definition (2.19), λj = i(2ωj + 2j Zj ),

j ∈ N.

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. [58] (for the case n = 3), [59, Sections 5, 6] (for n = 1, 2), and [106, Sections 5.2 and 7.4] (for n = 1 and n ≥ 3). For the practical purposes, in the case V ≡ 0, n ≤ 2, 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 (−Δ + V )Ψ = 0 which does not belong to L2 (see e.g. [59, Theorem 5.2 1 and 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] [58, Lemma 3.2]. Remark 2.4. We do not need to study the case λj → −2mi since the eigenvalues of the linearization at solitary waves are symmetric with respect to both real and imaginary axes; see e.g. [10]. 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 behavior 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 linearization 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 ∂ω Q(φω ) < 0, ω  m (see [68]). 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 families λNLS (ω) of the nonlinear Schrödinger equation with the same nonlinearity (linearized at a corresponding solitary waves). On the other hand, by [10, Theorem 2.19], there could be eigenvalue families of the linearization of the nonlinear Dirac operator which satisfy limω→m λa (ω) = ±2mi. Could these eigenvalues go off the imaginary axis into the complex plane? Theorem 2.3 states

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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 [12] 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) = φω,ξ (x)e−iωt + χω,η (x)eiωt ,

ξ, η ∈ CN/2 , |ξ|2 − |η|2 = 1,

(2.20)

where   v(r, ω)ξ φω,ξ (x) = i x · σ u(r, ω)ξ , r

 χω,η =

 −i xr · σ∗ u(r, ω)η , v(r, ω)η

(2.21)

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 [12]. 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 [12]. 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 [53,63,64,77]; see also [76] and [78, 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

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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 ∈ C 1 (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 that the threshold z = 1/(2m) of the operator l− is a regular point of its essential spectrum. Let φω (x)e−iωt , φω ∈ H 2 (Rn , CN ), ω  m, be the family of solitary waves constructed in Theorem 2.1. Then there is 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 ∂ω Q(φω ) < 0, which is formally in agreement with the Kolokolov stability criterion [68]. Proof. We consider the family of solitary wave solutions φω e−iωt , ω  m, described in Theorem 2.1. Let us assume that there is a sequence ωj → m and a family of eigenvalues λj of the linearization at solitary waves φωj e−iωj t such that Re λj = 0.   By [10, Theorem 2.19], the only possible accumulation points of the sequence λj j∈N are λ = ±2mi and λ = 0. By Theorem 2.3, as long as σd (l− ) = {0} and the threshold of l− is a regular point of the essential spectrum, λ = ±2mi can not be an accumulation point of nonzero-real-part eigenvalues; it remains to consider the case λj → λ = 0. By Theorem 2.2 (1), if Re λj = 0 and Λ0 is an accumulation point of the sequence Λj = λj /(m2 − ωj2 ), then Λ0 ∈ σp



0 −l+

l− 0



∩ R.

(2.22)

Above, l± (see (2.18)) correspond to the linearization at a solitary wave uκ (x)e−iωt with ω = −1/(2m) of the nonlinear Schrödinger equation (2.1). For κ ≤ 2/n, the spectrum of the linearization of the corresponding NLS at a solitary wave is purely imaginary:  σp

0 −l+

l− 0



⊂ iR.

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 ∂ω Q(φω ) > 0 for ω  m. On the other hand, as long

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as κ = 2/n and K > 4/n, Theorem 2.1 yields ∂ω Q(φω ) < 0 for ω  m, hence Λ0 = 0 would not be possible. We conclude that there is no family of eigenvalues (λj )j∈N with Re λj = 0. 2 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 [22, 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 essential spectrum of l− , with l− corresponding to the nonlinear Schrödinger equation in Rn (thus the spectral hypotheses of Theorem 2.2 and Theorem 2.3 are satisfied) as long as κ > κn ,

where κ1 = 1,

κ2 ≈ 0.621,

κ3 ≈ 0.461 .

Let us mention the related numerical results. In two dimensions, according to [22, Fig. 4], indeed l− does not have nonzero eigenvalues for κ > κ2 with κ2 ≈ 0.6. In three dimensions, according to [42, Fig. 3], l− does not have nonzero eigenvalues for κ > κ3 with some κ3 < 2/3. 3. Technical results 3.1. The linearization operator We assume that f ∈ C 1 (R \ {0}) ∩ C(R) satisfies Assumption 2.1 (recall Remark 2.2). Let φω (x)e−iωt be a solitary wave solution to equation (1.1) of the form (2.9), with ω ∈ (ω1 , m), where ω1 ∈ (m/2, m) is from Theorem 2.1. Consider the solution to (1.1) in the form of the Ansatz ψ(t, x) = (φω (x) + ρ(t, x))e−iωt , so that ρ(t, x) ∈ CN is a small perturbation of the solitary wave. The linearization at the solitary wave φω (x)e−iωt (the linearized equation on ρ) is given by i∂t ρ = L(ω)ρ,

L(ω) = Dm − ω − f (φ∗ω βφω )β − 2f  (φ∗ω βφω ) Re(φ∗ω β · )βφω . (3.1)

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 ) =

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O(|τ |κ−1 ) (see (2.16)) and the bound from below φ∗ω βφω ≥ |φω |2 /2 (see Theorem 2.1), the last term in (3.1) could be estimated by O(|φω |2κ ). Since L(ω) is not C-linear, in order to work with C-linear operators, we introduce the following self-adjoint matrices of size 2N × 2N :  αi =

Re αi Im αi



Re β β= Im β

 − Im αi , Re αi  − Im β , Re β

1 ≤ i ≤ n; 

0 J= −IN

 IN , 0

(3.2)

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) = Im φω (x) 

(3.3)

We mention that Jα · ∇x + βm is the operator which corresponds to Dm acting on R2N -valued functions. Introduce the operator L(ω) = Jα · ∇x + βm − ω − f (φ∗ω βφω )β − 2(φ∗ω β · )f  (φ∗ω βφω )βφω .

(3.4)

We extend this operator by C-linearity from L2 (Rn , R2N ) onto L2 (Rn , C2N ) = L2 (Rn , C ⊗R R2N ), with domain D(L(ω)) = H 1 (Rn , C2N ) where L(ω) is self-adjoint. The linearization at the solitary wave in (3.1) takes the form 

∂t ρ = JL(ω)ρ,

 Re ρ(t, x) ρ(t, x) = ∈ R2N , Im ρ(t, x)

(3.5)

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 [10] 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 {Jφω , ∂xi φω ; 1 ≤ i ≤ n} ⊂ ker(JL(ω)),

 1 Span Jφω , ∂ω φω , ∂xi φω , ωxi Jφω − αi φω ; 1 ≤ i ≤ n ⊂ L(JL(ω)). 2 The proof is in [10]. We mention the following relations (see e.g. [10]): JLJφω = 0,

JL∂ω φω = Jφω ;

JL∂xi φω = 0,

 1 JL ωxi Jφω − αi φω = ∂xi φω . (3.6) 2

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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 [38]. 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 L(ω) from (3.1) corresponding to the linearization at a (onefrequency) solitary wave has the eigenvalue −2ω of geometric multiplicity larger than or   equal to N/2, with the eigenspace containing the subspace Span χω,η ; η ∈ CN/2 , with χω,η defined in (2.21). The operator JL(ω) of the 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 −2ωχω,η = (−iα · ∇x + (m − f )β − ω)χω,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 . 2 3.2. Scaling, projections, and estimates on the effective potential Let πP = (1 + β)/2,

πA = (1 − β)/2,

π ± = (1 ∓ iJ)/2

(3.7)

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 πP± = π ± ◦ πP ,

± πA = π ± ◦ πA .

(3.8)

With ξ ∈ CN/2 , |ξ| = 1, from Theorem 2.1 (see (2.9)), we denote ⎤ Re ξ ⎢ 0 ⎥ ∈ R2N , Ξ=⎣ Im ξ ⎦ 0 ⎡

|Ξ| = 1.

(3.9)

For the future convenience, we introduce the orthogonal projection onto Ξ: ΠΞ = Ξ Ξ, · C2N ∈ End (C2N ). We note that, since βΞ = Ξ, one has

(3.10)

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Π Ξ ◦ π P = πP ◦ Π Ξ = Π Ξ ,

ΠΞ ◦ πA = πA ◦ ΠΞ = 0.

(3.11)

Denote by ψj ∈ H 1 (Rn , C2N ) eigenfunctions which correspond to the eigenvalues λj ∈ σp (JL(ωj )); thus, one has   L(ωj )ψj = Jα · ∇x + βm − ωj + v(ωj ) ψj = −Jλj ψj ,

j ∈ N,

(3.12)

where the potential v(ω) is defined by (cf. (3.4)) v(x, ω)ψ(x) = −f (φ∗ω βφω )βψ − 2φ∗ω βψ f  (φ∗ω βφω )βφω .

(3.13)

We will use the notations y = j x ∈ Rn ,  where j =

m2 − ωj2 , so that Jα · ∇x = j Jα · ∇y =: j D0 and Δ = 2j Δy . For v from

(3.13), define the potential V(y, ) ∈ End (C2N ) by V(y, ) = −2 v(−1 y, ω), −n/2

We define Ψj (y) = j

ω=

m2 −  2 ,

y ∈ Rn ,

 ∈ (0, 1 ).

(3.14)

ψj (−1 j y). With V from (3.14), L(ωj ) is given by

L(ωj ) = j D0 + βm − ωj + 2j V(ωj ),

j ∈ N,

(3.15)

and the relation (3.12) takes the form   j D0 + βm − ωj + Jλj + 2j V(ωj ) Ψj = 0,

j ∈ N.

(3.16)

We project (3.16) onto “particle” and “antiparticle” components and onto the ∓i spectral subspaces of J with the aid of projectors (3.7): − j D0 πA Ψj + (m − ωj − iλj )πP− Ψj + 2j πP− VΨj = 0,

(3.17)

− − Ψj + 2j πA VΨj = 0, j D0 πP− Ψj − (m + ωj + iλj )πA

(3.18)

+ Ψj + (m − ωj + iλj )πP+ Ψj + 2j πP+ VΨj = 0, j D 0 π A

(3.19)

j D0 πP+ Ψj

− (m + ωj −

+ iλj )πA Ψj

+

+ 2j πA VΨj

= 0.

(3.20)

We need several estimates on the potential V. Lemma 3.5. There is C > 0 such that for all y ∈ Rn and all  ∈ (0, 1 ) one has the following pointwise bounds:

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V(y, ) End (C2N ) ≤ C|uκ (y)|2κ , πP ◦ V(y, ) ◦ πA  End (C2N ) + πA ◦ V(y, ) ◦ πP  End (C2N ) ≤ C|uκ (y)|2κ ,   πA ◦ V(y, ) + |uκ (y)|2κ (1 + 2κΠΞ )β ◦ πA  End (C2N ) ≤ C2κ |uκ (y)|2κ ,   πP ◦ V(y, ) + |uκ (y)|2κ (1 + 2κΠΞ )β ◦ πP  End (C2N ) ≤ C2κ |uκ (y)|2κ . Above, uκ is the positive radially symmetric ground state of the nonlinear Schrödinger   equation (2.2); κ = min 1, K/κ − 1 > 0 was defined in (2.12). Proof. The bound on V(y, ) End (C2N ) follows from (2.15) and (2.16):   V(y, ) End (C2N ) ≤ C−2 |f (v 2 − u2 )| + v 2 |f  (v 2 − u2 )| ≤ C−2 v 2κ ≤ C|uκ (y)|2κ , where ω =

√

m2 − 2 ,  ∈ (0, 1 ), and |v(−1 |y|, ω)| ≤ C Vˆ (|y|) κ , 1

|u(−1 |y|, ω)| ≤ C Vˆ (|y|)1+ κ , 1

in the notations from Theorem 2.1, with Vˆ (|y|) = uκ (y) (see (2.6)). The bounds on πP ◦ V(y, ) ◦ πA  End (C2N ) and πA ◦ V(y, ) ◦ πP  End (C2N ) follow from πP ◦ V ◦ πA  End (C2N ) + πA ◦ V ◦ πP  End (C2N ) ≤ C|−2 f  (φ∗ω βφω )vu|, where |f  (φ∗ω βφω )| = |f  (v 2 − u2 )| ≤ C|v|2κ−2 by (2.8) and (2.16), with v, u bounded as above.   Let us prove the bound on πA ◦ V(y, ) + |uκ (y)|2κ (1 + 2κΠΞ )β ◦ πA  End (C2N ) . For ˆ , V˜ , U ˜ ∈ R which satisfy any numbers Vˆ > 0 and U 1 |U | ≤

V , 2

|V˜ | ≤ min

1 2

, C2κ Vˆ ,

(3.21)

ˆ +U ˜ , there are the following bounds: with V = Vˆ + V˜ and U = U  2  |f  κ (V 2 − 2 U 2 ) − 2 Vˆ 2κ |   2 ≤ |f  κ (V 2 − 2 U 2 ) − 2 (V 2 − 2 U 2 )κ | + 2 |(V 2 − 2 U 2 )κ − V 2κ | + 2 |V 2κ − Vˆ 2κ | ≤ c2K/κ (V 2 − 2 U 2 )K + O(2 V 2(κ−1) 2 U 2 ) + O(2 Vˆ 2κ−1 V˜ ) ≤ C2+2κ Vˆ 2κ ,

(3.22)

where we used (3.21) and also applied (2.15); similarly, using (2.16),  2  2 |f   κ (V 2 − 2 U 2 )  κ V 2 − κ2 Vˆ 2κ |   2   2 κ−1  2 2  κ V + κ2 |(V 2 − 2 U 2 )κ−1 V 2 − Vˆ 2κ | ≤ f   κ (V 2 − 2 U 2 ) − κ  κ (V 2 − 2 U 2 )

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2

2

≤ C| κ (V 2 − 2 U 2 )|K−1  κ V 2 + κ2 |(V 2 − 2 U 2 )κ−1 V 2 − V 2κ | + κ2 |V 2κ − Vˆ 2κ | ≤ C2+2κ Vˆ 2κ ; (3.23)  2 2  1+ 2 2 2  κ 2 2 2 2 2 κ−1 1+ 3 2κ |f  (V −  U )  κ U V | ≤ C| κ (V −  U )|  κ |U V | ≤ C Vˆ . (3.24) By (2.5), (2.11), and (2.10), we may assume that 1 > 0 in Theorem 2.1 is sufficiently small so that for  ∈ (0, 1 ) the functions V (t, ), U (t, ), Vˆ (t), V˜ (t, ) satisfy (3.21), pointwise in t ∈ R. Then, by (3.22), one has |−2 f − Vˆ 2κ | ≤ C2κ Vˆ 2κ ; so,     πA ◦ V + Vˆ 2κ (1 + 2κΠΞ )β ◦ πA  End (C2N ) ≤ πA ◦ V − Vˆ 2κ ◦ πA  End (C2N ) 2 ≤ C|−2 f − Vˆ 2κ | + C|−2 f  2+ κ U 2 | ≤ C2κ Vˆ 2κ ;

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∗ω βφω = V 2 −2 U 2 and are estimated with the aid of (2.15) and (2.16).   The bound on πP ◦ V(y, ) +|uκ (y)|2κ (1 +2κΠΞ )β ◦πP  End (C2N ) is derived similarly. We have:   πP ◦ V + Vˆ 2κ (1 + 2κΠΞ )β ◦ πP  End (C2N ) ≤  − −2 f − −2 2(φ∗ω πP · )f  πP φω + Vˆ 2κ (1 + 2κΠΞ ) End (C2N ) ≤ |−2 f − Vˆ 2κ | + |−2 2f  v 2 − Vˆ 2κ 2κ| ≤ C|−2 f − Vˆ 2κ | + C|−2 f  u2 | ≤ C2κ Vˆ 2κ . Above, we took into account that (φ∗ω βπP · )πP φω = ((πP βφω )∗ · )πP φω = ((πP φω )∗ · )πP φω = v 2 ΠΞ , with ΠΞ from (3.10) and v = v(|x|, ω) from (2.9), since   1 ξ (1 + β)φω (x) = v(|x|, ω) , 0 2

hence πP φω (x) = v(|x|, ω)Ξ.

2

4. Bifurcations from the origin Here we prove Theorem 2.2.   Lemma 4.1. Let ωj ∈ (0, m), j ∈ N; ωj → m. If there are eigenvalues λj ∈ σp JL(ωj )) , Re λj = 0, j ∈ N, such that lim λj = 0, then the sequence j→∞

Λj :=

λj , 2j

j∈N

does not have the accumulation point at infinity.

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s

iλj

−m−ωj

0

m−ωj

19

s

−iλj

Fig. 1. σ(Dm − ωj ) and ±iλj .

Proof. Due to the exponential decay of solitary waves stated in Theorem 2.1, there is C > 0 and s > 1/2 such that  r 2s V(·, ω)L∞ (Rn , End (C2N )) ≤ C,

∀ω ∈ (0, m).

(4.1)

Let Ψj ∈ L2 (Rn , C2N ), j ∈ N, be the eigenfunctions of JL(ωj ) corresponding to λj ; we then have (j D0 + βm − ωj + Jλj )Ψj = −2j V(ωj )Ψj ± (see (3.16)). Applying to this relation π ± = (1 ∓ iJ)/2 and denoting Ψ± j = π Ψj , we arrive at the system 2 + (j D0 + βm − ωj + iλj )Ψ+ j = −j π V(ωj )Ψj ,

(4.2)

2 − (j D0 + βm − ωj − iλj )Ψ− j = −j π V(ωj )Ψj .

Due to ωj → m, without loss of generality, we can assume that ωj > m/2 for all j ∈ N. Since the spectrum σ(JL) is symmetric with respect to real and imaginary axes, we may assume, without loss of generality, that Im λj ≥ 0 for all j ∈ N, so that Re(iλj ) ≤ 0 (see Fig. 1). Since λj → 0, we can also assume that |λj | ≤ m/2 for all j ∈ N. With j D0 + βm − ωj = Dm − ωj (considered in L2 (Rn , C ⊗R R2N )) being self-adjoint, one has (j D0 + βm − ωj − iλj )−1  =

1 1 = , dist(iλj , σ(Dm − ωj )) |m − ωj − iλj |

∀j ∈ N. (4.3)

From (4.2) and (4.3), using the bound (4.1) on V, we obtain Ψ− j  L2 ≤

2j 2j π − VΨj  ≤C  r −2s Ψj , |m − ωj − iλj | |m − ωj − iλj |

∀j ∈ N.

(4.4)

From (4.2) we have: 

m − ωj + iλj j D0

j D0 −m − ωj + iλj



  +  πP+ Ψj 2 πP VΨj = − , + + j πA Ψj πA VΨj

hence 

  πP+ Ψj m + ωj − iλj = + j D0 πA Ψj

with Δy = −D20 and

    −1 π + VΨj j D0 P , μj + Δy + −(m − ωj + iλj ) πA VΨj

(4.5)

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μj =

(−m − ωj + iλj )(m − ωj + iλj ) , 2j

j ∈ N;

(4.6)

we note that Im μj = −2−2 j (ωj + Im λj ) Re λj = 0 for all j ∈ N, so that μj + Δy is invertible indeed. We may assume that inf j∈N |μj | > 0, or else there would be nothing to prove: if μj → 0, we would have |λj − i(m − ωj )| = o(2j ), hence |λj | = O(2j ). Then, by the limiting absorption principle (cf. Lemma A.1),  r −s π + Ψj  ≤ C|μj |−1/2  r s π + VΨj  + Cj  r s π + VΨj . The above, together with (4.4) and the bound (4.1) on V, leads to −s +  r −s Ψj  ≤  r −s Ψ− Ψj  ≤ C j  +  r



2j 1 −s + Ψj . 1 + j  r

|m − ωj − iλj | |μj | 2

If we had |λj |/2j → ∞, then |m − ωj − iλj | ≥ |λj |/2 for j large enough, hence |μj | ≥ m|m − ωj − iλj |/2j ≥ m|λj |/(22j ) for j large enough (since ωj → m and λj → 0 in (4.6)),  r −s Ψj  ≤ C

 2 j j −s + +  Ψj . j  r

|λj | |λj | 12

Due to |λj |/2j → ∞, the above relation would lead to a contradiction since Ψj ≡ 0 for all j ∈ N. We conclude that the sequence Λj = λj /2j can not have an accumulation point at infinity. 2 Lemma 4.2. For any η ∈ C \ R+ there is s0 (η) ∈ (0, 1), lower semicontinuous in η, such that the resolvent (−Δ − η)−1 defines a continuous mapping (−Δ − η)−1 : L2s (Rn ) → Hs2 (Rn ),

0 ≤ s < s0 (η).

Proof. Let f ∈ L2s (Rn ); define u = (−Δ − η)−1 f ∈ H 2 (Rn ). There is the identity (−Δ − η)( r s u) + [ r s , −Δ]u = r s (−Δ − η)u,

(4.7)

which holds in the sense of distributions. Taking into account that [ r s , −Δ]u ≤ CuH 1 O(s) ≤ Cf L2s O(s), one concludes from (4.7) that (−Δ − η)( r s u) ∈ L2 (Rn ) and hence r s u ∈ L2 (Rn ), both being bounded by Cf L2s , with some C = C(η) < ∞, thus so is uHs2 . 2

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21

It is convenient to introduce the following operator acting in L2 (Rn , C2N ): K=

Δ 1 − − u2κ κ (1 + 2κΠΞ ), 2m 2m

D(K) = H 2 (Rn , C2N ).

(4.8)

Above, ΠΞ ∈ End (C2N ) is the orthogonal projector onto Ξ ∈ R2N ; see (3.9), (3.10). Lemma 4.3.  ⎧  0 l− ⎪ ⎪ , ⎨σ −l+ 0   1. σ(JK|R(πP ) ) =  ⎪ 0 l− ⎪ ⎩σ ∪ σ(il− ) ∪ σ(−il− ), −l+ 0 The equality also holds for the point spectra. 2. One has:  dim L(JK|R(πP ) ) =

2n + N,

N = 2; N ≥ 4.

κ = 2/n;

2n + N + 2,

(4.9)

κ = 2/n.

Above, K is from (4.8) and l± were introduced in (2.18). Proof. We decompose L2 (Rn , R(πP )) into the direct sum X1 ⊕ X2 , where    X1 = L2 Rn , Span Ξ, JΞ) ,    ⊥  X2 = L2 Rn , Span Ξ, JΞ ∩ R(πP ) .

(4.10)

Note that both Ξ and JΞ belong to R(πP ). The proof of Part 1 follows once we notice that JK is invariant in the spaces X1 and X2 , and that JK|X1 is rep    0 l− resented in L2 Rn , Span Ξ, JΞ by , while JK|X2 is represented in −l+ 0      ⊥  0 l− L2 Rn , Span Ξ, JΞ ∩ R(πP ) by I N −1 ⊗C . We also notice that if −l− 0 2 N = 2, then X2 = {0}. The proof of Part 2 also follows from the above decomposition and the relations  dim L(JK|X1 ) = dim L

0 −l+

l− 0

 =

 2n + 2,

κ = 2/n;

2n + 4,

κ = 2/n

(cf. Lemma B.3) and dim L(JK|X2 ) = (N − 2) dim L(l− ) = (N − 2) dim ker(l− ) = N − 2.   Remark 4.4. JK|R(πA ) is represented in L2 Rn , R(πA ) by IN/2 ⊗C



0 −l−

 l− . 0

2

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22

Since σ(JL) is symmetric with respect to real and imaginary axes (see Remark 2.4), we assume without loss of generality that λj satisfies Im λj ≥ 0,

∀j ∈ N.

(4.11)

Passing to a subsequence, we assume that Λj =

λj → Λ0 ∈ C. 2j

(4.12)

Lemma 4.5. 1. If Λ0 ∈ / σ(JK), then there is C > 0 such that 2κ κ πP Ψj  + −1 j πA Ψj  ≤ Cj uκ Ψj ,

∀j ∈ N.

2. For s > 0 sufficiently small, there is C > 0 such that − κ πP− Ψj Hs1 + −1 j πA Ψj Hs1 ≤ Cuκ Ψj ,

∀j ∈ N.

3. If Λ0 ∈ / i[1/(2m), +∞), then + κ πP+ Ψj Hs1 + −1 j πA Ψj Hs1 ≤ Cuκ Ψj ,

∀j ∈ N.

4. If Λ0 ∈ i[1/(2m), +∞), then κ + κ uκκ πP+ Ψj  + −1 j uκ πA Ψj  ≤ Cuκ Ψj ,

∀j ∈ N.

Proof. Let us prove Part 1. We divide (3.17), (3.19) by 2j and (3.18), (3.20) by j , arriving at 

1 m+ωj

+ JΛj D0

which we rewrite as  1 m+ωj

D0 −m − ωj + 2j JΛj



πP Ψ j −1 j πA Ψj



 =−

 πP V(y, j )Ψj , j πA V(y, j )Ψj

  πP Ψ j − u2κ D0 κ (1 + 2κΠΞ ) + JΛj −1 D0 −m − ωj j πA Ψj   π (V(y, j ) + u2κ κ (1 + 2κΠΞ ))Ψj . =− P j πA (V(y, j ) + JΛj )Ψj

The Schur complement of the entry −m − ωj is given by Tj =

1 Δy + Λj J − u2κ . κ (1 + 2κΠΞ ) − m + ωj m + ωj

(4.13)

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23

If Λj → Λ0 ∈ / σ(JK), then Tj has a bounded inverse for j sufficiently large, and then the operator in the 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 

± iΛj D0

1 m+ωj

D0 −m − ωj



πP± Ψj −1 ±  j π A Ψj



 =−

 πP± V(y, j )Ψj . ± j πA (V(y, j ) ± iΛj )Ψj

(4.14)

Denote the matrix-valued operator in the left-hand side of (4.14) by A± j . The Schur complement of (A± ) is given by j 22 ± ± −1 ± Tj± = (A± j )11 − (Aj )12 (Aj )22 (Aj )21 =

1 Δy ± iΛj − . m + ωj m + ωj

(4.15)

Since Im Λ0 ≥ 0 (cf. (4.11)), Tj− is invertible in L2 (except perhaps at finitely many − values of j which we disregard); writing the inverse of A− j in terms of Tj , we conclude − from (4.14) that πP− Ψj H 1 + −1 j πA Ψj H 1 ≤ CVΨj . Moreover, by Lemma 4.2, for sufficiently small s > 0, − κ πP− Ψj Hs1 + −1 j πA Ψj Hs1 ≤ CVΨj L2s ≤ Cuκ Ψj ,

j ∈ N.

This proves Part 2. As long as Λ0 ∈ / i[1/(2m), +∞), Part 3 is proved in the same way as Part 2. To prove Part 4, we write 

1 m+ωj

+ iΛj + μu2κ κ D0

D0 −m − ωj



  +  πP+ Ψj πP (V − μu2κ κ )Ψj , = − + + −1 j πA (V + iΛj )Ψj j π A Ψj

(4.16)

with some μ ≥ 0 to be specified. The Schur complement of −m − ωj is Tj =

1 Δy + iΛj + μu2κ . κ − m + ωj m + ωj

We pick μ ≥ 0 such that the threshold z = 1/(2m) is a regular point of the essential 1 Δ spectrum of the operator 2m + μu2κ κ − 2m , which is by [59] a generic situation; indeed, by (C.1), the virtual levels correspond to the situation when M = −1 + μK is not invertible, with K a compact operator. (For n ≥ 3, it is enough to take μ = 0 since −Δ has no virtual level at z = 0; for n ≤ 2, it is enough to take μ > 0 small [90].) Then, by Lemma C.1, uκκ ◦ Tj−1 ◦ uκκ (for j large enough) is bounded in L2 , and the conclusion follows from (4.16). Above, uκκ ◦ . . . ◦ uκκ denotes the compositions with the operators of multiplication by uκκ . 2 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,

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there is the inclusion Λ0 ∈ σ(JK). Proposition 4.6. If Λ0 = limj→∞ λj /2j and Re λj = 0 for all j ∈ N, then Λ0 ∈ σp (JK) ∩ R. Proof. From now on, we assume that the corresponding eigenfunctions Ψj (cf. (3.16)) are normalized: Ψj 2 = 1,

j ∈ N.

(4.17)

2 By (4.13), −1 j D0 πA Ψj is bounded in L uniformly in j ∈ N, while by Lemma 4.5 (2), κ −1 (3) and (4) so is uκ j πA Ψj . Again by (4.13), uκκ D0 πP Ψj is uniformly bounded in L2 . −1 1 n 2N It follows that both −1 ) and contain weakly j πP Ψj and j πA Ψj belong to Hloc (R , C convergent subsequences; we denote their limits by

ˆ ∈ H 1 (Rn , C2N ), P loc

ˆ ∈ H 1 (Rn , C2N ). A loc

(4.18)

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   2κ ˆ P D0 2m − uκ (1 + 2κΠΞ ) + JΛ0 (4.19) ˆ = 0. D0 −2m A  Let us argue that if Re λj = 0 for all j ∈ N, then

ˆ P ˆ A

 is not identically zero. By

Lemma 4.5 (2), using the compactness of the Sobolev embedding Hs1 ⊂ L2 , we conclude that there is an infinite subsequence (which we again enumerate by j ∈ N) such that ˆ ∈ H 1 (Rn , C2N ), πP− Ψj → π − P

− −ˆ 1 n 2N −1 ) j πA Ψj → π A ∈ H (R , C

(4.20)

as j → ∞, with the strong convergence in L2 . Remark 4.7. If additionally Λ0 ∈ / i[1/(2m), +∞), then Tj+ from (4.15) is also invertible; just like above, one concludes that there is an infinite subsequence (which we again enumerate by j ∈ N) such that ˆ ∈ H 1 (Rn , C2N ), πP+ Ψj → π + P

+ +ˆ 1 n 2N −1 ) j πA Ψj → π A ∈ H (R , C

(4.21)

as j → ∞, with the strong convergence in L2 . Lemma 4.8. Let J be skew-symmetric and let L be symmetric linear operators in a Hilbert space X , with J 2 = −IX . If λ ∈ σp (JL) \iR and Ψ ∈ X is a corresponding eigenvector, then Ψ, LΨ = 0 and Ψ, JΨ = 0.

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25

Proof. If Re λ = 0, then both sides of the identity Ψ, LΨ = −λ Ψ, JΨ equal zero since Ψ, LΨ ∈ R, Ψ, JΨ ∈ iR. 2 ˆ − = 0. Lemma 4.9. If Re λj = 0 for all j ∈ N, then P − 2 2 Proof. Lemma 4.8 yields 0 = Ψj , JΨj = iΨ+ j  − iΨj  , for all j ∈ N; thus, by (4.17), − 2 2 2 Ψ+ j  = Ψj  = Ψj  /2 = 1/2,

∀j ∈ N.

(4.22)

Therefore,   ˆ − 2 = lim π − Ψj 2 = lim π − Ψj 2 + π − Ψj 2 = 1/2, P P P A j→∞

j→∞

∀j ∈ N.

(4.23)

Above, in the first two relations, we took into account (4.20). 2 

 ˆ P Thus, ˆ ∈ L2 (Rn , C2N ×C2N ) is not identically zero, hence Λ0 ∈ σp (JK). It remains A to prove that Λ0 ∈ R.

(4.24)

Let us assume that, on the contrary, Λ0 ∈ σp (JK) ∩ (iR \ {0}). By (4.11), it is enough to consider Λ0 = ia,

a > 0.

(4.25)

ˆ − 2 = 1/2. Since By (4.23), P   ˆ + 2 ≤ lim π + Ψj 2 ≤ lim π + Ψj 2 + π + Ψj 2 = 1/2, P P P A j→∞

j→∞

we arrive at the inequality ˆ + 2 − P ˆ − 2 ≤ 0. P

(4.26)

ˆ = iaP ˆ it follows that From the above and from JKP ˆ KP

ˆ = P, ˆ −iaJP

ˆ = a P ˆ +, P ˆ + − a P ˆ −, P ˆ − ≤ 0. P,

(4.27)

Remark 4.10. If Λ0 belongs to the spectral gap of JK (Λ0 ∈ iR, |Λ0 | < 1/(2m)), then both ± ±ˆ 1 n 2N πP± Ψj and −1 ) j πA Ψj (up to choosing a subsequence) converge to π P ∈ H (R , C ±ˆ 1 n 2N 2 and π A ∈ H (R , C ) strongly in L (cf. Remark 4.7). Then, by the above arguments, ˆ ± 2 = 1/2 and hence P, ˆ JP

ˆ = 0 = P, ˆ KP . ˆ P

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26

Lemma 4.11. If Λ0 ∈ iR \ {0}, Λ0 ∈ σp



0 −l+

l− 0



, and z ∈ L2 (Rn , C2 ) is a corre-

sponding eigenfunction, then   l z, + 0

  0 z > 0. l−

  0 l− Proof. Let z be an eigenfunction which corresponds to Λ0 ∈ σp ∩ iR, −l+ 0   p Λ0 = 0. Let p, q ∈ L2 (Rn , C) be such that z = and let Λ0 = ia with a ∈ R\{0}. Then iq      p 0 l− p = ia results in ap = l− q and aq = l+ p (note that q ∈ / ker(l− ); iq −l+ 0 iq otherwise one would conclude that p ≡ 0 and then also 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 , hence    p l , + iq 0

0 l−

  p = p, l+ p + q, l− q = 2 q, l− q > 0, iq

where we took into account that l− is semi-positive-definite and that q ∈ / ker(l− ).

2

Since K is invariant where  in the subspaces X1 and X2 defined in (4.10),   it is reprel+ 0 l− 0 and by a positive-definite operator IN/2−1 ⊗C , respectively sented by 0 l− 0 l− (see the proof of Lemma 4.3), it follows from Lemma 4.11 that the quadratic form · , K · = · , K · |X1 + · , K · |X2 = · , K · |X1 + · , (IN −2 ⊗ l− ) · |X2 is strictly positive-definite on any eigenspace of JK corresponding to Λ0 = ia ∈ σp (JK), a > 0. Therefore, ˆ KP

ˆ > 0. P,

(4.28)

The relations (4.27) and (4.28) lead to a contradiction; we conclude that (4.24) is satisfied. 2 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(2 ).

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27

Proposition 4.12. There is δ > 0 sufficiently small and 2 > 0 such that ∂Dδ2 ⊂ ρ(JL) for all  ∈ (0, 2 ), and for the Riesz projector Pδ,

1 =− 2πi





JL(ω) − η

−1

dη,

ω=



m2 −  2

(4.29)

|η|=δ2

one has rank Pδ, = 2n + N if κ = 2/n, and 2n + N + 2 otherwise. One also has dim ker(JL(ω)) = n + N − 1. Remark 4.13. Let us first give an informal calculation of rank Pδ, , 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 Kolokolov condition ∂ω Q(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 translational symmetry [7]. Proof of Proposition 4.12. Let δ > 0 be such that Dδ ∩ σ



0 −l+

l− 0



= {0}.

Let us define the operator L (ω) = −2 L(ω) = −1 D0 + −2 (βm − ω) + V(y, ω),

ω ∈ (ω1 , m)

(4.31)

√ (cf. (3.15)), where y = x,  = m2 − ω 2 , and D0 is the Dirac operator in the variables y = x (we recall that D0 = Jα · ∇y = Jα · ∇x ). We rewrite (4.29) as follows: Pδ, = −

1 2πi

 |η|=δ

(JL (ω) − η)−1 dη,

ω=

m2 −  2 .

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Lemma 4.14. Let 1 pδ = − 2πi



(JK − η)−1 πP dη

|η|=δ

be the Riesz projector onto the generalized null space of JK|R(πP ) . Then: 1.   πP Pδ, πP  π P π A δ, P

  πP Pδ, πA p − δ πA Pδ, πA 0

 0  →0 0 L2 (Rn ,C4N )→L2 (Rn ,C4N )

as  → 0;

2. There is 2 > 0 such that, for any  ∈ (0, 2 ), one has rank Pδ, = rank pδ . Proof. By Lemma 4.3, σ(JK) ⊂ σ



0 −l+

l− 0



∪σ(il− ) ∪σ(−il− ), hence (JK −η)|R(πP )

has a bounded inverse (JK − η)−1 : H −1 (Rn , R(πP )) → H 1 (Rn , R(πP )) on the circle |η| = δ, η ∈ C, with δ > 0 sufficiently small (cf. Lemma B.4). On the direct sum (R(πP )) ⊕ (R(πA )), the operator JL (ω) − η is represented by the matrix     A11 (, η) A12 () πP J L π A π JL π P − η := P . A21 () A22 (, η) π A JL π P π A JL π A − η According to (4.31), A12 ()H 1 →L2 + A12 ()L2 →H −1 + A21 ()H 1 →L2 = O(−1 ), A22 (, η)−1 |R(πA ) = −

2 −1 J + OL2 →L2 (4 ). 2m

(4.32)

In the last equality, we used the following relation (cf. (4.31)): A22 (, η) = πA JL πA − η = −−2 (m + ω)J + πA JV(y, ω)πA − η. The Schur complement of A22 (, η) is given by T (, η) = A11 (, η) − A12 ()A22 (, η)−1 A21 () (4.33)  J + JV − η πP − πP (−1 JD0 + JV)πA A22 (, η)−1 πA (−1 JD0 + JV)πP , = πP m+ω which we consider as an operator T (, η) : H 1 (Rn , C2N ) → H −1 (Rn , C2N ). With the expression (4.32) for A22 (, η)−1 , the Schur complement (4.33) takes the form

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 T (, η) = πP

JΔy J + JV − η − + OH 1 →H −1 (2 ) πP . m+ω 2m

29

(4.34)

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(πP ) ⊕ R(πA )) → L2 (Rn , R(πP ) ⊕ R(πA )), as follows (see (D.6)): −1

(JL − η)



T −1 = −1 −A−1 22 A21 T

 T −1 A12 A−1 22 . −1 −1 A−1 A12 A−1 22 + A22 A21 T 22

(4.35)

Above, T = T (, η), A12 = A12 (), A21 = A21 (), A22 = A22 (, η). Since    T (, η) − (JK − η) |R(πP ) H 1 →H −1 = O(),

(4.36)

uniformly in |η| = δ, while JK − η : H 1 (Rn , C2N ) → H −1 (Rn , C2N ) has a bounded inverse for |η| = δ, the operator T (, η)|R(πP ) is also invertible for |η| = δ as long as  > 0 is sufficiently small, with its inverse being a bounded map from H −1 (Rn , R(πP )) to H 1 (Rn , R(πP )). Using (4.32), we conclude that the matrix (4.35) has all its entries, except the top left one, of order O() (when considered in the L2 → L2 operator norm). Hence, it follows from (4.35) and (4.36) that, considering Pδ, as an operator on R(πP ) ⊕ R(πA ),   p  Pδ, − 0δ

   0   1 =  0 2πi



T (, η)−1 − (JK − η)−1 0

   0 dη  + O() = O(), 0

|η|=δ

where the norms refer to L2 → L2 operator norm. This proves Lemma 4.14 (1). The statement (2) follows since both Pδ, and pδ are projectors. 2 The statement of Proposition 4.12 on the rank of Pδ, 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 Pˆδ, = − (L (ω) − η)−1 dη, ω = m2 −  2 , 2πi |η|=δ

similarly to how it was done for Pδ, , and from the relation ker(JL (ω)) = ker(L (ω)) = R(Pˆδ, ), Above, δ > 0 is small enough so that

 ∈ (0, 2 ).

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Dδ ∩ σ



0 l−

l+ 0



= {0}.

This finishes the proof of Proposition 4.12. 2 Now we return to the proof of Theorem 2.2 (2). If there is an eigenvalue family (λj )j∈N , λj λj ∈ σp (JL(ωj )), such that λj = 0 for all j ∈ N, Λj = m2 −ω 2 → 0 as ωj → m, then j the dimension of the generalized kernel of the nonrelativistic limit of the rescaled system   jumps up, so that dim L JL(ω) |ω
e1 (y, ω) = − κ Jφω (−1 y), 1

e2 (y, ω) = 

1 2− κ

(∂ω φω )(

(4.37)

−1

y);

√ here and below,  = m2 − ω 2 . Noting the factor −2 in the definition of L in (4.31), we deduce from (3.6) the relations JL (ω)e1 (ω) = 0,

JL (ω)e2 (ω) = e1 (ω),

ω ∈ (ω1 , m).

(4.38)

With θ(y) = −

m uκ (y) − m y · ∇uκ (y) κ

θ ∈ H 1 (Rn )

(4.39)

l+ β(y) = α(y)

(4.40)

and real-valued α, β ∈ H 2 (Rn ) such that l+ θ(y) = uκ (y),

l− α(y) = θ(y),

(see (B.7), (B.8), and (B.10) in the proof of Lemma B.3), we define E3 (y) = −JΞα(y),

E4 (y) = −Ξβ(y),

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with Ξ ∈ R2N from (3.9), so that E3 , E4 ∈ H 2 (Rn , R2N ) satisfy JKE3 (y) = Ξθ(y), Lemma 4.15. Let ω2 =

JKE4 (y) = E3 (y).

(4.41)

m2 − 22 , with 2 from Proposition 4.12. The functions

ea (ω), 1 ≤ a ≤ 2,

e3 (ω) = JL(ω)e4 (ω),

e4 (ω) = Pδ, E4 ,

ω ∈ (ω2 , m),

can be extended to continuous maps ea : (ω2 , m] → L2 (Rn , R2N ), 1 ≤ a ≤ 4, with e1 (m) = JΞuκ , e2 (m) = Ξθ, and ea (m) = limω→m ea (ω) = Ea , 3 ≤ a ≤ 4, so that JKe1 (ω) = 0,

JKe2 (ω) = e1 (ω),

JKe3 (m) = e2 (m),

ω ∈ (ω2 , m];

JKe4 (m) = e3 (m).

(4.42)

Proof. By Theorem 2.1, e1 (y, ω) = − κ Jφω (−1 y) = JΞuκ (y) + OH 1 (Rn ,C2N ) (2κ ), 1

so lim e1 (y, ω) is defined in H 1 (Rn , C2N ). Since ω→m

ˆ (r) + U ˜ (r, )), v(r, ω) = 1/κ (Vˆ (r) + V˜ (r, )) and u(r, ω) = 1+1/κ (U ˆ from (2.6), one has ∂ω v(x, ω) = with Vˆ , U

∂ ∂ω ∂



 1 1  κ Vˆ (x) +  κ V˜ (x, ) , so that

2− κ ∂ω v(−1 y, ω)  Vˆ (y) V˜ (y, ) + y · ∇Vˆ (y) + + y · ∇V˜ (y, ) + ∂ V˜ (y, ) . = −ω κ κ 1

(4.43)

Using (2.11) from Theorem 2.1 to bound the y · ∇V˜ -term, one has |y|∇y V˜ (|y|, )L2 (Rn ) = O(2κ ); due to (2.13) from Theorem 2.1, ∂ V˜ (·, )H 1 (Rn ,R2 ) = O(2κ−1 ). Taking into account these estimates in (4.43), we arrive at 2− κ (∂ω v)(−1 y, ) = −ω 1

1 κ

Vˆ (y) + y · ∇Vˆ (y) + OL2 (Rn ) (2κ ),

1

with a similar expression for 2− κ ∂ω u. This leads to 2− κ (∂ω φω )(−1 y) = −ω 1

1 κ

Vˆ (y) + y · ∇Vˆ (y) Ξ + OL2 (Rn ) (2κ ).

(4.44)

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Taking into account that e2 (y, ω) = 2− κ (∂ω φω )(−1 y) (cf. (4.37)), the relation (4.44) allows us to define 1

e2 (m) := lim e2 (ω) = lim 2− κ (∂ω φω )(−1 · ) = Ξθ, 1

ω→m

(4.45)

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, E4 = E4 + lim (Pδ, − pδ )E4 = E4 ,

ω→m

ω→m

ω→m

with the limit holding in L2 norm. In the last relation, we used the relation pδ Ea = Ea , 1 ≤ a ≤ 4, and Lemma 4.14. For a = 3, the result follows from e3 (ω) = JL(ω)e4 (ω) = JL(ω)Pδ,ω e4 (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 E4 ∈ L2 (Rn , R2N ) is real-valued, while Pδ,ω commutes with the operator K : C2N → C2N of complex conjugation since JL has real coefficients. 2 In the vector space R(Pδ, ) we may choose the basis (cf. Lemma 3.2) 

ea (ω), 1 ≤ a ≤ 4;

 ∂yi Φ, ωy i JΦ − αi Φ, 1 ≤ i ≤ n; 2

 Θk , 1 ≤ k ≤ N − 2 , (4.46)

where Φ is from (4.37) and Θk (ω) are certain vectors from ker(JL (ω)), with 1 ≤ k ≤ N − 2 due to Proposition 4.12 (which states that rank Pδ, = 2n + N + 2, dim ker(JL (ω)|Pδ, ) = n + N − 1). 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 [7], the span of these vectors, span{Θk ; 1 ≤ k ≤ 3}, turns out to contain the null eigenvector e1 (ω). In the basis (4.46) of the space R(Pδ, ), the operator (JL (ω) − λIN )|R(Pδ, ) is represented by ⎡

Mω − λIN

−λ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ =⎢ ⎢ ... ⎢ ⎢ . ⎢ .. ⎣ .. .

1 −λ 0 0 .. . .. . .. .

σ1 (ω) σ2 (ω) σ3 (ω) − λ σ4 (ω) .. . .. . .. .

0 0 0 0 1 0 −λ 0 .. . −λIn .. . 0 .. . 0

0 0 0 0

0 0 0 0

In

0

−λIn

0

0

−λIN −2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦

(4.47)

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33

where vertical dots denote columns of irrelevant coefficients and σa (ω),

1 ≤ a ≤ 4,

are some continuous functions of ω. We used Lemma 4.15 and the relations   JL ωy i JΦ − αi Φ = ∂yi Φ, 1 ≤ i ≤ n, 2 which follow from (3.6), (4.31), and (4.37). Considering (4.47) at λ = 0 and  = 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(Mω − λ) = (−λ)2n+N (λ2 − λσ3 (ω) − σ4 (ω)).

(4.49)

1 Lemma 4.17. For any solitary wave φ(x)e−iωt with φ ∈ H1/2 (Rn ) and any 1 ≤ i ≤ n, one has φ, αi φ = 0.

Proof. The local version of the charge conservation, ∂μ J μ = 0, with J μ (t, x) = ¯ x)γ μ ψ(t, x), when applied to a solitary wave with stationary charge and current ψ(t, μ ¯ densities, J μ (t, x) = φ(x)γ φ(x), yields the desired identity: for any 1 ≤ i ≤ n,

J (x)x dx = − 0

0 = ∂t

i

n

 

 ∂j J (x) xi dx =

j=1Rn

Rn

J i (x) dx.

j

2

Rn

Expanding JL e3 (ω) over the basis in R(Pδ, ) (see (4.46)), we conclude that for some continuous functions γi (ω) and ρi (ω), 1 ≤ i ≤ n, and τk (ω), 1 ≤ k ≤ N − 2, there is a relation JL e 3 =

4  a=1

σa ea +

n   i=1

−2 N   τk Θ k , γi ∂yi Φ + ρi ωy i JΦ − αi Φ + 2

(4.50)

k=1

valid for ω ∈ (ω2 , m]. Above, σa (ω), 1 ≤ a ≤ 4, are continuous functions from (4.47). Pairing (4.50) with Φ(ω) = J−1 e1 (ω), we get: ! ! 0 = σ2 (ω) J−1 e1 (ω), e2 (ω) + σ4 (ω) J−1 e1 (ω), e4 (ω) ,

ω ∈ (ω2 , m].

(4.51)

We took into account that one has Φ, v = L e2 , v = e2 , L v = 0 for any v ∈ ker(JL ), the identities ! J−1 e1 , JL e3 (ω) = − L e1 , e3 (ω) = 0,

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J−1 e1 , e3 = J−1 e1 , JL e4 = − L e1 , e4 = 0, and also the identity Φ, ωxi JΦ − 2 αi Φ = 0 which holds due to Lemma 4.17 and due to Φ∗ JΦ ≡ 0 (the left-hand side is skew-adjoint while all the quantities are real-valued). Since J−1 e1 (ω), e2 (ω) = − κ φω (−1 ·), (∂ω φω )(−1 ·)

2

2

= n− κ φω , ∂ω φω = ∂ω Q(φω )/2

(4.52)

(we took into account that κ = 2/n), the relation (4.51) takes the form σ2 (ω)∂ω Q(φω )/2 = μ(ω)σ4 (ω),

ω ∈ (ω2 , m],

(4.53)

where μ(ω) := − J−1 e1 (ω), e4 (ω) is a continuous function of ω ∈ (ω2 , m]. Remark 4.18. By (4.52) and Lemma 4.15, ∂ω Q(φω ) is a continuous function of ω ∈ (ω2 , m]. Lemma 4.19. There is ω3 ∈ (ω2 , m) such that μ(ω) > 0 for ω3 < ω ≤ m. Proof. We have μ(ω) = − Φ, e4 = − Φ, Pδ, (e4 ) = − J−1 e1 (m), e4 (m) + O(), while (4.42) yields ! − J−1 e1 , e4 |ω=m = − Ke2 , e4 |ω=m = − JKe3 , J−1 e3 |ω=m = Ξα, KΞα > 0. Above, we used (4.41) and the explicit form of E2 and E3 .

2

Lemma 4.20. There is ω4 ∈ [ω3 , m) such that σ3 (ω) ≡ 0 for ω ∈ [ω4 , m]. Proof. Applying (JL (ω))2 to (4.50), we get (JL )3 e3 (ω) = σ3 (ω)(JL )2 e3 (ω) + σ4 (ω)(JL )2 e4 (ω). Coupling this relation with J−1 e4 and using the identities J−1 e4 , (JL )3 e3 = e3 , L JL e3 = 0 and J−1 e4 , (JL )2 e4 = − e4 , L JL e4 = 0

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(both of these due to skew-adjointness of L JL , taking into account that ea (ω), 1 ≤ a ≤ 4, are real-valued by Lemma 4.15, while J and L have real coefficients), we have σ3 (ω) J−1 e4 , (JL )2 e3 = 0.

(4.54)

The factor at σ3 (ω) is nonzero for ω < m sufficiently close to m. Indeed, using (4.48), J−1 e4 , (JL )2 e3 |ω=m = J−1 e4 , σ2 e1 + σ3 JL e3 + σ4 e3 |ω=m = J−1 e4 , e1 |ω=m = − e4 , φ |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 σ3 (ω) is identically zero for ω ∈ [ω4 , m], with some ω4 < m. 2 Since σ3 (ω) is identically zero for ω ∈ [ω4 , m], we conclude from (4.49) that the nonzero eigenvalues of JL (ω) satisfy λ2 − σ4 (ω) = 0, ω ∈ [ω4 , m]. By (4.48) and Lemma 4.19, the relation (4.53) shows that σ4 (ω) is of the same sign as ∂ω Q(φω ). Thus, if ∂ω Q(φω ) > 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 ∼ ∂ω Q(φω ) for ω  m; hence, there are two real eigenvalues of JL, of magnitude ∼ 2 ∂ω Q(φω ). This completes the proof of Theorem 2.2. 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), Zj = −

2ωj + iλj ∈ C, 2j

j ∈ N,

(5.1)

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 spectrum of l− (neither an L2 -eigenvalue nor a virtual level). Lemma 5.1. There is C > 0 such that uκκ π + Ψj L2 ≤ Cj uκκ Ψj L2 ,

∀j ∈ N.

Proof. The relations (3.19) and (3.20) yield 

m − ωj + iλj j D0

j D0 −(m + ωj − iλj )



  +  πP+ Ψj 2 πP VΨj = − , + + j πA Ψj πA VΨj

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hence 

  πP+ Ψj m + ωj − iλj = + j D0 πA Ψj

    −1 π + VΨj j D0 P , Δy + ζj2 + −(m − ωj + iλj ) πA VΨj

(5.2)

with   ζj2 := (ωj − iλj )2 − m2 /2j = (8m2 + o(1))/2j ,

Re ζj ≥ 0;

j ∈ N.

(Note that since ωj → m, Re λj = 0, and λj → 2mi, one has Im ζj2 = 0 for all but finitely many j ∈ N which we discard.) The limiting absorption principle from Lemma A.1 with ν = 0, 1 and with z = ζj2 shows that there is C > 0 such that uκκ ◦ (Δy + ζj2 )−1 ◦ uκκ  ≤ C|ζj |−1 , uκκ ◦ j D0 (Δy + ζj2 )−1 ◦ uκκ  ≤ Cj ,

∀j ∈ N,

(5.3)

∀j ∈ N,

(5.4)

where uκκ ◦ . . . ◦ uκκ denotes the compositions with the operators of multiplication by uκκ . Applying (5.3) and (5.4) to (5.2) leads to   uκκ π + Ψj  ≤ C uκκ D0 (Δy + ζj2 )−1 π + VΨj  + uκκ (Δy + ζj2 )−1 π + VΨj    ≤ C uκκ ◦ D0 (Δy + ζj2 )−1 ◦ uκκ  + uκκ ◦ (Δy + ζj2 )−1 ◦ uκκ  uκκ Ψj  ≤ Cj uκκ Ψj ,

∀j ∈ N.

We used the bound V(y, ) End (C2N ) ≤ C|uκ (y)|2κ from Lemma 3.5.

2

Lemma 5.2. The values Zj , j ∈ N from (5.1) are uniformly bounded: There is C > 0 such that |Zj | ≤ C,

∀j ∈ N.

  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: |(ωj + iλj )2 − m2 | < 2j .

(5.5)

This implies that (ωj + iλj )2 = m2 + O(2j ), hence −ωj − iλj = m + O(2j ) (since λj → −2mi as ωj → m); taking into account the relation 2j Zj = −2ωj − iλj (see (5.1)), we arrive at

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2j Zj = m − ωj + O(2j ) = O(2j ), which shows that |Zj | are uniformly bounded for the values of j for which (5.5) takes place. Now we discard these values of j ∈ N; assuming that there are infinitely many left (or else there is nothing to prove), we have: |(ωj + iλj )2 − m2 | ≥ 2j ,

∀j ∈ N.

We write (3.17), (3.18) as the following system:   −   −  m − ωj − iλj j D0 πP Ψj 2 πP VΨj = −j , − − j D0 −(m + ωj + iλj ) πA Ψj πA VΨj

(5.6)

j ∈ N,

which can then be rewritten as follows:  −     −1  − π P Ψj m + ωj + iλj j D0 πP VΨj = , + ν Δ y j − − j D0 −(m − ωj − iλj ) πA Ψj πA VΨj

(5.7)

(5.8)

with νj :=

(ωj + iλj )2 − m2 , 2j

j ∈ N.

(5.9)

We notice that |νj | ≥ 1 by (5.6) and that Im νj = 0 except perhaps for finitely many values of j, which we discard. Applying (5.3) and (5.4) to (5.8), we derive:   uκκ π − Ψj  ≤ C j + |νj |−1/2 uκκ Ψj ,

∀j ∈ N.

(5.10)

By Lemma 5.1 and (5.10), there is C > 0 such that   uκκ Ψj  ≤ uκκ π + Ψj  + uκκ π − Ψj  ≤ C j + |νj |−1/2 uκκ Ψj , ∀j ∈ N.

(5.11)

Assume that lim sup |νj | = +∞. Then the coefficient at uκκ Ψj  in the right-hand side j→∞

of (5.11) would go to zero for an infinite subsequence of j → ∞; since Ψj ≡ 0, we arrive at the contradiction. Thus, |νj | are uniformly bounded. From (5.9), we derive:  ωj + iλj = − m2 + 2j νj = −m + O(2j ), where we chose the “positive” branch of the square root (for all but finitely many j ∈ N) since ωj → m and λj → 2mi as j → ∞. This yields   2 − 2 − m m2 + 2j νj j 2ωj + iλj ωj + (ωj + iλj ) Zj = − = − = − , j ∈ N; 2j 2j 2j therefore, Zj = O(1) are uniformly bounded and could not accumulate at infinity. 2

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m − ωj − iλj D0

m+ωj +iλj 2j

Zj −

=−

2ωj +iλj 2j



D0

1 m+ωj

+ u2κ κ

−

m−ωj 2j

= Zj −

1 m+ωj ,

we rewrite (5.7) as

   2j πP− VΨj πP− Ψj =− . (5.12) − − − j πA Ψj j πA VΨj − j u2κ κ π A Ψj

We denote the matrix-valued operator in the left-hand side by   D0 m − ωj − iλj Aj := D Z − 1 + u2κ , 0

Aj : L2 (Rn , C4N ) → L2 (Rn , C4N ),

j

m+ωj

(5.13)

κ

D(Aj ) = H 1 (Rn , C4N );

j ∈ N.

We note that (Aj )11 → 2m as j → ∞, hence, for j large enough, (Aj )11 is invertible. By (D.4), the Schur complement of (Aj )11 , j ∈ N, is given by  Sj = (Aj )22 − (Aj )21 (Aj )−1 (A ) = Zj − j 12 11

1 Δy + u2κ + I2N , κ m + ωj m − ωj − iλj (5.14)

with Sj : L2 (Rn , C2N ) → L2 (Rn , C2N ), D(Sj ) = H 2 (Rn , C2N ) for j ∈ N.   Let Z0 be the limit point of the sequence Zj j∈N ; by Lemma 5.2, Z0 = ∞. Lemma 5.3. Z0 ∈ σd (l− ) ∪ {1/(2m)}. If, moreover, the threshold z = 1/(2m) is a regular point of the essential spectrum of l− , then Z0 ∈ σd (l− ). Proof. We write the inverse of the matrix-valued operator in the left-hand side of (5.12) using (D.4): "

#

πP− Ψj − j πA Ψj

1 =− (Aj )11

"

−1 1 + (Aj )−1 11 D0 Sj D0

−D0 Sj−1

−Sj−1 D0

(Aj )11 Sj−1

#"

# 2j πP− VΨj .   − j πA V − u2κ κ Ψj (5.15)

Above, the Schur complement of (Aj )11 = m − ωj − iλj is given by the expression (5.14): Sj = hj I2N ,

j ∈ N,

with hj : L2 (Rn ) → L2 (Rn ) (with domain D(hj ) = H 2 (Rn )) given by hj = Zj −

1 Δy + u2κ κ + m + ωj m − ωj − iλj

1 1 1 − − m − ωj − iλj m + ωj m − ωj − iλj  2m 2mu2κ Δy κ + 1− + u2κ κ + m − ωj − iλj m − ωj − iλj m − ωj − iλj  2m 2m =− (l− − Zˆj ) + 1 − u2κ , m − ωj − iλj m − ωj − iλj κ = Zj +

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where the sequence m − ωj − iλj  1 1 Zˆj = − , Zj + 2m m − ωj − iλj m + ωj

j ∈ N,

  has the same limit as the sequence Zj j∈N . Let Ω ⊂ C be a compact set as in Lemma C.1: Ω ∩ σd (l− ) = ∅, and if z = 1/(2m) is an eigenvalue or a virtual level of l− , then we also assume that 1/(2m) ∈ / Ω. Then, by Lemma C.1, there is C > 0 such that uκκ ◦ (l− − z)−1 ◦ uκκ L2 →H 2 ≤ C,

∀z ∈ Ω \ [1/(2m), +∞).

Since Z0 ∈ Ω, one has Zˆj ∈ Ω and also Zˆj ∈ / σ(l− ) for j ∈ N large enough. We note that (m − ωj − iλj )2 m − ωj − iλj  2ωj + iλj 1 1 1 − ; Zˆj = + − = 2 2 2m m − ωj m − ω − iλj m − ωj 2m 2m(m2 − ωj2 ) (m−ωj +Im λj ) Re λj since Im λj → 2m and Re λj > 0, we conclude that Im Zˆj = = 0 (hence 2m(m2 −ωj2 ) ˆ / σ(l− )) for all j ∈ N except at most for finitely many values of j which we discard. Zj ∈

Then the mapping uκκ ◦ Sj−1 ◦ uκκ : L2 (Rn ) → H 2 (Rn ),

j ∈ N,

is bounded uniformly in j ∈ N. By duality, so is the mapping H −2 (Rn ) → L2 (Rn ), and, by complex interpolation, so is uκκ ◦ Sj−1 ◦ uκκ : H −1 (Rn ) → H 1 (Rn ). It follows that the following maps are also continuous: uκκ ◦ Sj−1 ◦ D0 ◦ uκκ : L2 (Rn ) → H 1 (Rn ), uκκ ◦ D0 ◦ Sj−1 ◦ uκκ : H −1 (Rn ) → L2 (Rn ), uκκ ◦ D0 ◦ Sj−1 ◦ D0 ◦ uκκ :

L2 (Rn ) → L2 (Rn ),

with the bounds which are uniform in j ∈ N. We recall that there is the inclusion uκ ∈ C 2 (Rn ) (see (2.3)), with both uκ and its derivatives exponentially decaying with |x| ([11, Lemma A.1]); moreover, inf x∈Rn |∂r u(x)/u(x)| > 0 ([11, Lemma A.2]). As a consequence, for any κ > 0 (with κ < 2/(n − 2) if n ≥ 3), the multiplication by uκκ extends to a continuous mapping in L2 (Rn ), in H 1 (Rn ), and by duality in H −1 (Rn ). Similarly, the matrix multiplication by [D0 , uκκ ] = κuκ−1 Jαi ∂xi uκ extends to a continuous mapping in L2 (Rn , C2N ). κ Therefore, if either Zj → Z0 ∈ C \ σ(l− ) or Zj → Z0 ∈ (1/(2m), +∞) (with Z0 = 1/(2m) if z = 1/(2m) either an eigenvalue or a virtual level of l− ), with Im Zj = 0, then,

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taking into account that λj → 2mi as ωj → m, we conclude that the relation (5.15) yields the following inequality:  1  −  1 − −  πP VΨj  + Cj  κ πA VΨj − u2κ κ π A Ψj κ uκ uκ  1  1   1  −   − − ≤ C2j  κ πP− VΨj  + Cj  κ πA VπP− Ψj + κ πA (V − u2κ )π Ψ (5.16) κ A j , uκ uκ uκ

− uκκ πP− Ψj  + j uκκ πA Ψj  ≤ C2j 

with C = C(Z0 ) > 0, and then, using the bounds V End (C2N ) ≤ C|uκ (y)|2κ ,

πA ◦ V ◦ πP  End (C2N ) ≤ C|uκ (y)|2κ ,

and − − 2κ 2κ πA (V + u2κ κ (1 + 2κΠΞ )β)πA  End (C2N ) ≤ C |uκ (y)|

from Lemma 3.5 and taking into account the identity − − − − 2κ πA (V − u2κ κ )πA = πA (V + uκ (1 + 2κΠΞ )β)πA ,

we obtain the estimate uκκ π − Ψj  ≤ Cj

min(1,2κ)

uκκ Ψj ,

∀j ∈ N.

(5.17)

 min(1,2κ)  κ The inequality (5.17) and Lemma 5.1 lead to uκκ Ψj  = O j uκ Ψj , in contradiction to Ψj ≡ 0, j ∈ N. Thus, the assumption Z0 ∈ C \ σ(l− ) leads to a contradiction, and so does the assumption Z0 ∈ (1/(2m), +∞), and so does the assumption Z0 = 1/(2m) when the threshold z = 1/(2m) is a regular point of the essential spectrum of l− . 2 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 = 2ωj i 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 2 ∈ (0, 1 ) such that for any  ∈ (0, 2 ) and any z ∈ D1 the relations D0 πA Y − (ω − iλ − m)πP Y + 2 πP+ V(X + Y) = 0, + D0 πP Y − (ω − iλ + m)πA Y + 2 πA V(X + Y) = 0,

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where ω =

√

41

m2 − 2 and λ = λ(z) = (2ω + 2 z)i

(5.18)

(cf. (2.19)), define a linear map ϑ(·, , z) : L2,−κ (Rn , R(π − )) → L2,−κ (Rn , R(π + )), ϑ(·, , z) : X → Y, which is analytic in z, where for μ ∈ R the exponentially weighted spaces are defined by   L2,μ (Rn ) = u ∈ L2loc (Rn ); eμr u ∈ L2 (Rn ) ,

uL2,μ := eμr uL2 .

Moreover, there is C > 0 such that ϑ(·, , z)L2,−κ (Rn ,C2N )→L2,−κ (Rn ,C2N ) ≤ C,

 ∈ (0, 2 ),

z ∈ D1 ,

∂z ϑ(·, , z)L2,−κ (Rn ,C2N )→L2,−κ (Rn ,C2N ) ≤ C ,

 ∈ (0, 2 ),

z ∈ D1 .

3

Proof. By (5.2), 

 %−1  +   $ (ω − iλ)2 − m2 πP Y πP V(X + Y) ω − iλ + m D0 . I = Δy + + πA Y D0 ω − iλ − m V(X + Y) πA 2

This leads to  (ω − iλ)2 − m2 −1 Y = π + {(ω − iλ) + m(πP − πA ) + D0 } Δy + I V(X + Y). (5.19) 2 For  > 0 and z ∈ C, Im z < 0 (so that Re λ(z) > 0 by (5.18)), we define the linear map Φ(·, , z) : L2,−κ (Rn , C2N ) → L2,−κ (Rn , R(π + )), (5.20)  2 2 −1 (ω − iλ) − m Φ(Ψ, , z) = π + {(ω − iλ) + m(πP − πA ) + D0 } Δy + I VΨ, 2 where λ = λ(z) = (2ω + 2 z)i. Using the definition (5.20), the relation (5.19) takes the form Y = Φ(X + Y, , z).

(5.21)

  Since the norm of Φ(·, , z) ∈ B L2,−κ (Rn , C2N ), L2,−κ (Rn , C2N ) , z ∈ D1 , is small (as long as  > 0 is small enough), we will be able to use the above relation to express Y as a function of X ∈ L2,−κ (Rn , R(π − )). −m Remark 5.5. The inverse of Δy + (ω−iλ) is not continuous in λ at Re λ = 0; this 2 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. 2

2

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In view of Proposition A.2, it is convenient to change the variables so that in (5.20) we deal with (−Δy − ζ 2 ); let ζ ∈ C be defined by ζ 2 = ((ω − iλ)2 − m2 )/2 ,

Re ζ ≥ 0;

(5.22)

the values of λ with Re λ > 0 correspond to the values of ζ with Im ζ < 0 (since λ ∈ D2 (2ωi) and ω ∈ (ω2 , m), where ω2 = m2 − 22 , with 2 ∈ (0, 1 ) small enough). Due to Lemma 3.5, V(y, ) End (C2N ) ≤ Ce−2κ|y| ; using the analytic continuation of the resolvent from Proposition A.2, 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 (−Δy − ζ 2 )−1 for Im ζ < 0 (this corresponds to Re λ > 0) and its analytic continuation 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(2 ).

(5.24)

Therefore, by (5.22), Re ζ = −1 Re

(ω − iλ)2 − m2 = O(−1 ),

(5.25)

showing that for  sufficiently small one has ζ ∈ C \ D1 (we take 2 > 0 smaller if necessary). Since we only consider z ∈ D1 , the relation (5.18) yields | Re λ| ≤ |z|2 ≤ 2 , and then (5.22) and (5.25) lead to Im ζ =

1 Im((ω − iλ)2 − m2 ) O(Re λ) Im ζ 2 = O() = O() = O(), 2 2 Re ζ  2

showing that the condition (5.23) holds true for  sufficiently small, satisfying assumptions of Proposition A.2. Then, by Proposition A.2, there is C > 0 such that for any Ψ ∈ L2,−κ (Rn , C2N ) the map (5.20) satisfies Φ(Ψ, , z))L2,−κ (Rn ,C2N ) ≤ CΨL2,−κ (Rn ,C2N ) ,

 ∈ (0, 2 ),

z ∈ D1 .

(5.26)

We take 2 smaller if necessary so that 2 ≤ 1/(2C) (with C > 0 from (5.26)); then the linear map I − Φ(·, , z) : Y → Y − Φ(Y, , z) is invertible, with (I − Φ(·, , z))−1 L2,−κ →L2,−κ ≤ 2,

 ∈ (0, 2 ),

z ∈ D1 .

(5.27)

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Since Φ(·, , z) is linear, writing (5.21) in the form Y − Φ(Y) = Φ(X), we can express Y = (I − Φ)−1 Φ(X). Thus, for each  ∈ (0, 2 ) and z ∈ D1 , we may define the mapping (X, , z) → Y, which we denote by ϑ: ϑ(·, , z) : L2,−κ (Rn , R(π − )) → L2,−κ (Rn , R(π + )), ϑ(·, , z) : X → Y = (I − Φ(·, , z))−1 Φ(X, , z).

(5.28)

By (5.26) and (5.27), one has ϑ(·, , z)L2,−κ →L2,−κ ≤ 2C, for  ∈ (0, 2 ) and z ∈ D1 . 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     = 2|ζ∂z ζ| = ∂z ∂ (2ω +  z) z    = 2|ω − iλ|, 2 2

(5.29)

with the right-hand side bounded uniformly in  ∈ (0, 2 ) and z ∈ D1 . Therefore, using the bound for the derivative of the analytic continuation of the resolvent (cf. Proposition A.2, which we apply with ν = 0 and also with ν = 1 to accommodate the operator D0 from the definition of Φ in (5.20)), we conclude that there is C > 0 such that ∂z Φ(·, , z)L2,−κ →L2,−κ ≤ ∂ζ ΦL2,−κ →L2,−κ |∂z ζ| ≤

C |ω − iλ| ≤ C3 ζ 2 |ζ|

for all  ∈ (0, 2 ) and z ∈ D1 . Above, ∂z is considered as a gradient in R2 ∼ = C; we used (5.29) and the estimate (5.25). Then it follows from (5.28) that there is C > 0 such that one also has ∂z ϑ(·, , z)L2,−κ →L2,−κ ≤ C3 , for all  ∈ (0, 2 ) and z ∈ D1 . One can see from (5.20) that Φ is analytic in the complex parameter z, hence so is ϑ. 2 As long as j ∈ N is sufficiently large so that j ∈ (0, 2 ) and zj ∈ D1 , then, by Lemma 5.4, the relations (3.19) and (3.20) allow us to express π + Ψj = ϑ(π − Ψj , j , zj ), with ϑ(·, , z) a linear map from L2,−κ (Rn , C2N ) into itself, with ϑ(·, , z)L2,−κ (Rn ,C2N )→L2,−κ (Rn ,C2N ) ≤ C. Now (3.17) and (3.18) can be written as

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− j D0 πA Ψj + (m − ωj − iλj )πP− Ψj + 2j πP− V ϑ π − Ψj = 0,

(5.30)

− Ψj + 2j πP− V ϑ π − Ψj = 0, j D0 πP− Ψj − (m + ωj + iλj )πA

(5.31)

where V ϑ = V ϑ (y, j , λj ), with V ϑ (y, , z) := V(y, ) ◦ (1 + ϑ(·, , z)) satisfying   V ϑ (, λ)L2 →L2 ≤ V()L2,−κ →L2 1 + ϑ(·, , z)L2,−κ →L2,−κ ≤ C, so that   V(j )Ψj = V(j )(π − Ψj + π + Ψj ) = V(j ) π − Ψj + ϑ(π − Ψj , j , zj ) = V ϑ π − Ψj . We recall the definition Zj = −(2ωj + iλj )/2j (cf. (5.1)) and rewrite (5.30), (5.31), as the following system: "

πP− (

m+ωj + Zj + V ϑ )πP− 2j − −1 πA (j D0 + V ϑ )πP−

ϑ − πP− (−1 j D0 + V )πA − πA (Zj −

1 m+ωj

− + V ϑ )πA

#

 πP− Ψj = 0, − πA Ψj

(5.32)

with j ∈ N. We rewrite the above as the nonlinear eigenvalue problem 



 πP− Ψ A(, z) − = 0, πA Ψ

+ z + V ϑ )πP− π − ( m+ω 2 A(, z) := P− −1 πA ( D0 + V ϑ )πP−

−  πP− (−1 D0 + V ϑ )πA − − , 1 πA (z − m+ω + V ϑ )πA

(5.33)

where the operator A(, z) : H 1 (Rn , R(π − )) → L2 (Rn , R(π − )) is considered as     − − A(, z) : L2 Rn , R(πP− ) × R(πA ) → L2 Rn , R(πP− ) × R(πA ) , − with domain D(A(, z)) = H 1 (Rn , R(πP− ) × R(πA )). Note that the operator A(, z) depends on z analytically via ϑ (cf. Lemma 5.4). By Weyl’s theorem,

 & '  σess (A(, z)) = − ∞, −(m + ω)−1 + z ∪ (m − ω)−1 + z, +∞ ,

(5.34)

so that 0 ∈ / σess (A(, z)). Thus, the values Zj defined in (5.1) are such that the kernel of A(j , z) is nontrivial at z = Zj ; such values of z are called the characteristic roots (or, informally, nonlinear eigenvalues) of A(, z). Nonlinear eigenvalue problem. We will study the location of characteristic roots of A(, z) using the theory developed by M. Keldysh [63,64]; see also [53,77]. 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) [62, 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

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that for any u, v ∈ D, each z0 ∈ Ω, and each η in the resolvent set of A(z0 ), the function u, (A(z) − η)−1 v is analytic in z in an open neighborhood of z0 and that A(z) is resolvent-continuous in z (in the norm topology). Definition 5.6. The point z0 ∈ Ω is said to be regular for the operator-valued analytic function A(z) if the operator A(z0 ) has a bounded inverse. If the equation A(z0 )ϕ = 0 has a non-trivial solution ϕ0 ∈ D, then z0 is said to be a characteristic root of A and ϕ0 an eigenvector of A corresponding to z0 . Assumption 5.7. z0 ∈ Ω is normal characteristic root of A(z). This means that A(z) has a bounded inverse for 0 < |z − z0 | < R, with some R > 0, and 0 ∈ σd (A(z0 )); see [63,64]. Due to Assumption 5.7, there is δ > 0 such that Dδ ∩ σ(A(z0 )) = {0}. Due to the resolvent continuity of A in z, there is an open neighborhood U ⊂ Ω of z0 such that ∂Dδ ⊂ ρ(A(z)) for all z ∈ U . Let Pδ,z = −

1 2πi



(A(z) − η)−1 dη,

z ∈ U.

|η|=δ

One has rank Pδ,z = rank Pδ,z0 < ∞ (the last inequality due to the assumption that 0 ∈ σd (A(z0 ))). Definition 5.8. The multiplicity α ∈ N of the characteristic root z0 of A(z) is the order of vanishing of det A(z)|R(Pδ,z ) at z0 . Remark 5.9. The above definition does not depend on a particular choice of δ > 0. Lemma 5.10. Let z0 be a normal characteristic root of A(z) of multiplicity α ∈ N. The geometric multiplicity of 0 ∈ σd (A(z0 )) satisfies g ≤ α. Proof. We denote ν = dim R(Pδ,z0 ) < ∞ and choose the basis {ψi }1≤i≤ν in R(Pδ,z0 ) so that ψi , 1 ≤ i ≤ g, are eigenvectors corresponding to zero eigenvalue of A(z0 ). Let A(z) be the matrix representation of A(z) in the basis {Pδ,z ψi }1≤i≤ν . Then the first g columns of A(z) vanish at z = z0 , hence det A(z) = O((z − z0 )g ). 2 The sum of multiplicities of characteristic roots is stable under perturbations (cf. [53, Theorem 2.1]): Theorem 5.1. Let A(, z) : X → X,  ∈ R, z ∈ Ω, be a continuous family of operators with domains D(A(, z)) = D, ∀(, z) ∈ R × Ω, which is resolvent-continuous in  and z and analytic in z ∈ Ω. Let z0 be a normal characteristic root of A(0, z) of multiplicity

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α ∈ N. There is an open neighborhood Ω ⊂ C of z0 such that if  ≥ 0 is sufficiently small, then the sum of multiplicities of all characteristic values of A(, z) inside Ω equals α. Proof. Fix δ > 0 sufficiently small so that σ(A(0, z0 )) ∩ Dδ = {0}. Due to the resolvent continuity, there is 1 > 0 and an open neighborhood Ω ⊂ C of z0 such that for all  ∈ [0, 1 ] and all z ∈ Ω one has σ(A(, z)) ∩ Dδ = {0}. Let Γ = ∂Dδ and denote PΓ (, z) = −

1 2πi



(A(, z) − ηI)−1 dη,

 ∈ [0, 1 ],

z ∈ Ω.

Γ

Due to the resolvent continuity of A(, z) in  and z, one has rank PΓ (, z) = rank PΓ (0, z0 ) =: ν,

∀ ∈ [0, 1],

∀z ∈ Ω.

  Let ψi 1≤i≤ν be the basis in R(PΓ (0, z0 )). Denote ψi (, z) = PΓ (, z)ψi ,

1 ≤ i ≤ ν,

 ∈ [0, 1],

z ∈ Ω;

it is a basis in R(PΓ (, z)) (we take Ω and 1 > 0 smaller if necessary). Let M(, z) be the matrix representation of A(, z), restricted onto R(PΓ (, z)), in this basis. If  ∈ [0, 1 ] and zj ∈ Ω is a characteristic root of A(, z), then its multiplicity equals the order of vanishing of det M(, z) at zj . Now the proof follows from Cauchy’s argument principle applied to det M(, z) in Ω when we take Ω and 1 > 0 small enough so that zeros of det M(, z) do not intersect ∂Ω. 2 Now we apply the above theory to the operator A(, 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 σess (A(, z)) ∩ D1/(4m) = ∅

∀ω ∈ (ω∗ , m),

∀ ∈ (0, ∗ ),

∀z ∈ U,

where ∗ = m2 − ω∗2 . Since 0 ∈ σd (l− ), with l− from (2.18), we may assume that the open neighborhood U  {0} is small enough so that U ∩ σ(l− ) = {0}.

(5.35)

Let Aij (, z), i, j = which are the entries of A(, z) in (5.33),  1, 2, denote the operators  A11 (, z) A12 (, z) so that A(, z) = . We then have: A21 (, z) A22 (, z) A12 H 1 →L2 + A21 H 1 →L2 + A21 L2 →H −1 = O(−1 ), A11 L2 →L2 = O(−2 ),

(5.36)

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and, taking ∗ > 0 smaller if necessary, one has A−1 11 =

2 + OL2 →L2 (4 ), 2m

∀ ∈ (0, ∗ ),

∀z ∈ U,

(5.37)

with the estimate OL2 →L2 (4 ) uniform in z ∈ U . This suggests that we study the invertibility of A(, z) in terms of the Schur complement of A11 (, z), which is defined by − − 1 n −1 S(, z) = A22 − A21 A−1 (Rn , R(πA )), 11 A12 : H (R , R(πA )) → H

(5.38)

where A11 , A12 , A21 , and A22 are evaluated at (, z) ∈ (0, ∗ ) × U . Let us derive the explicit expression for S(, z):  − S(, z) = πA z−

 − 1 + V ϑ πA m+ω  −1 − − − − πA (D0 + V ϑ )πP− m + ω + 2 (V ϑ + z) πP (D0 + V ϑ )πA  ϑ  − 1 Δy  − − + u2κ πA + π A V − u2κ κ + κ πA m+ω m+ω 2 D 1 0 − − − − πA π − (D0 + V ϑ )πA + πA − πA (D0 + V ϑ )πP− m+ω m+ω P 2 (V ϑ + z) −1 1  − − 1+ − πA (D0 + V ϑ )πP− − 1 πP− (D0 + V ϑ )πA . m+ω m+ω

 − = πA z−

Above, uκ = uκ (x) is the ground state of the nonlinear Schrödinger equation (2.2). We note that, by (3.13), − ϑ − − − − 2κ πA V πA = πA V ◦ (1 + ϑ)πA = πA uκ + OL2 →L2 (); 1

we used the bounds πA φω L∞ = O(1+ κ ) (cf. Theorem 2.1) and ϑL2−s →L2−s = O() (cf. Lemma 5.4) which yield V ◦ϑL2 →L2 ≤ VL2−s →L2s ϑL2−s →L2−s = O() for s > 1/2. Thus, taking into account Lemma B.4, the operator S(, z) defined in (5.38) takes the form  Δy 1 − − + u2κ + OH 1 →H −1 () πA S(, z) = πA ,  ∈ [0, ∗ ), (5.39) z− κ + 2m 2m with the estimate OH 1 →H −1 () uniform in z ∈ U . Above, we extended S(, z) in (5.39) from  ∈ (0, ∗ ) to  ∈ [0, ∗ ) by continuity. The following lemma allows us to reduce the problem of studying the characteristic roots of A(, z) (see (5.33)) to the characteristic roots of S(, z) (cf. (5.38)). Lemma 5.11. If ∗ > 0 is sufficiently small, then for all  ∈ [0, ∗ ) the point z0 ∈ D1/(2m) is a characteristic root of A(, z) if and only if it is a characteristic root of S(, z).

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Proof. Since the operator A11 (, z) : L2 (Rn , R(πP− )) → L2 (Rn , R(πP− )) is invertible for  > 0 small enough (see (5.37)), the expression for the Schur complement (D.4) and the relation (D.2) shows that the operator A(, z) from (5.33) is invertible if and only if so is S(, z). 2 Lemma 5.12. The multiplicity of the characteristic root z0 = 0 of S(0, z) is α = N/2. − . Since dim ker(l− ) = 1, one has dim ker(S(0, 0)) Proof. By (5.39), S(0, 0) = (z − l− )πA − − 2N ); then = rank πA = N/2. Let ei ∈ C , 1 ≤ i ≤ N/2, be the basis in R(πA   uκ ei 1≤i≤N/2 is the basis in ker(S(0, 0)). We do not need to use the Riesz projectors since the operator S(0, z) is invariant in this space, being represented by S(0, z) = zIN/2 ; thus, det S(0, z) = z N/2 . 2

Lemma 5.13. There is no sequence of characteristic roots Zj = 0 of A(j , z) such that Zj → 0 as j → ∞. Proof. Let δ > 0 be such that ∂Dδ ⊂ ρ(A(, z0 )), z0 = 0. Due to the continuity of the resolvent in z and , there is ∗ ∈ (0, 2 ) and an open neighborhood U ⊂ D1/(2m) , z0 ∈ U , such that ∂Dδ ⊂ ρ(A(, z)) for all z ∈ U and  ∈ [0, ∗ ). 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(, z) for all  ∈ [0, ∗ ). At the same time, by Lemma 3.4 and Lemma 5.10, z = 0 is a characteristic root of S(, z),  ∈ [0, ∗ ), of multiplicity at least α = N/2. Hence, there can be no other, nonzero characteristic roots z ∈ U of S(, z) for any  ∈ [0, ∗ ), and, in particular, given a sequence j → 0, there is no sequence of characteristic roots Zj of S(j , z) such that Zj = 0 for j ∈ N, Zj → 0 as j → ∞. By Lemma 5.11, the same conclusion holds true for A(, z). 2 By Lemma 5.13, Zj = 0 for all but finitely many j ∈ N. By (2.19), this implies that λj = 2ωj i for all but finitely many j ∈ N. This concludes the proof of Theorem 2.3 (3). The proof of Theorem 2.3 is now complete. Appendix A. Analytic continuation of the free resolvent Let us remind the limiting absorption principle for the resolvent of the free Laplacian. Lemma A.1 (Limiting absorption principle for the Laplace operator). Let n ≥ 1. For any s > 1/2, δ > 0, and ν ∈ N0 , ν ≤ 2, there is C = C(n, s, δ, ν) such that −(1−ν)/2 ν (Rn ) ≤ C|z| (−Δ − z)−1 L2s (Rn )→H−s ,

∀z ∈ C \ (Dδ ∪ R+ ).

For the proof, see [1, Theorem A.1 and Remark 2 in Appendix A].

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N. Boussaïd, A. Comech / Journal of Functional Analysis ••• (••••) ••••••

49

Let Eμ : L2 (Rn ) → L2 (Rn ) denote the operator of multiplication by e−μr , μ ∈ R. Following [86], not to confuse the regularized resolvent Rμ0 (ζ 2 ) := Eμ R0 (ζ 2 )Eμ = Eμ (−Δ − ζ 2 )−1 Eμ ,

ζ ∈ C,

Im ζ > 0

with its analytic continuation through the line Im ζ = 0, we will denote the latter by Fμ0 (ζ). Proposition A.2 (Analytic continuation of the resolvent). Let n ≥ 1, μ > 0. 1. There is an analytic function Fμ0 (ζ), Fμ0 : {Im ζ > −μ} \ (−iR+ ) −→ B (L2 (Rn ), L2 (Rn )), such that Fμ0 (ζ) = Rμ0 (ζ 2 ) for Im ζ > 0, and for any δ > 0 and ν ∈ N0 , ν ≤ 2, there is C = C(n, μ, δ, ν) < ∞ such that Fμ0 (ζ)L2 →H ν ≤

C , (1 + |ζ|)1−ν

Im ζ ≥ −μ + δ,

dist(ζ, −iR+ ) > δ.

(A.1)

2. If n is odd and satisfies n ≥ 3, then (A.1) holds for all ζ ∈ C, Im ζ ≥ −μ + δ. Remark A.3. This result in dimension n = 3 was stated and proved in [86, Proposition 3], as a consequence of the explicit expression for the integral kernel of Rμ0 (ζ 2 ), −

e−μy eiζ|y−x| e−μx , 4π|y − x|

Im ζ > 0,

x, y ∈ R3 ,

which could be extended analytically to the region Im ζ > −μ as a holomorphic function of ζ with values in L2 (R3 ×R3 ). In [86], 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 [86] since we consider the resolvent of −Δ instead of Δ.) Proof. Let us define the analytic continuation of Fμ0 (ζ). For u, v ∈ L2 (Rn ) we define uμ , vμ ∈ L2,μ (Rn ) by uμ (x) = e−μx u(x), vμ (x) = e−μx v(x) and consider

I(ζ) =

v, Fμ0 (ζ)u

v( μ (ξ)

= Rn

1 dn ξ u ( (ξ) , μ ξ2 − ζ 2 (2π)n

(A.2)

which is an analytic function in ζ ∈ C+ := {z ∈ C : Im z > 0}. 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

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N. Boussaïd, A. Comech / Journal of Functional Analysis ••• (••••) •••••• Im λ 6

q

0

q

a−δ

q

q

a

a+δ

a−2δ

q

-Re λ

a+2δ δ Ka

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

by

the

contour

γa

Kaδ = {ζ ∈ C ; a − δ ≤ Re ζ ≤ a + δ, −μ + δ ≤ Im ζ ≤ 0}

=

(A.3)

(see Fig. 2), satisfying there the bounds (A.1) with constants cj independent of a. We pick a > 0 and δ > 0, with a ≥ 3δ, and break the integral (A.2) into two: I(ζ) =

(δ) I1 (ζ)

+

(δ) I2 (ζ)

=

+

||ξ|−a|>2δ

.

(A.4)

||ξ|−a|<2δ

The first integral in (A.4) is finite, being bounded by

(δ)

|I1 (ζ)| ≤

|ˆ vμ (ξ)||ˆ uμ (ξ)| ||ξ|−a|>2δ



≤ Rn

  1  dn ξ 1  1 − 2|ζ|  |ξ| − ζ |ξ| + ζ  (2π)n

uμ vμ  |ˆ vμ (ξ)||ˆ uμ (ξ)| 2 dn ξ , ≤ n 2|ζ| δ (2π) |ζ|δ

and therefore is analytic in ζ and is bounded by C/|ζ|. Above, to estimate the denominators, we took into account that for ζ ∈ Kaδ and ||ξ| − a| > 2δ, ||ξ| ± ζ| ≥ |(|ξ| − a) + (a ± Re ζ)| ≥ ||ξ| − a| − |a ± Re ζ| > 2δ − δ = δ. To analyze the second integral in (A.4), we will deform the contour of integration in ξ. ∞ Let g0 ∈ Ccomp (R) be even, g0 ≤ 0, supp g0 ∈ [−2δ, 2δ], with g0 (0) = −μ + δ/2 and monotonically increasing to zero as |t| → 2δ. Moreover, we may assume that |g0 | < 4μ/δ and that dist(γ0 , K0δ ) ≥ δ/2, where Kaδ is defined in (A.3) and γ0 = {(λ, g0 (λ)) : |λ| ≤ 2δ}; see Fig. 2. Define ga (t) = g0 (t − a). Lemma A.4. Let u ∈ L2,μ (Rn ), so that uL2,μ (Rn ) := eμr uL2 (Rn ) < ∞. Then its Fourier transform, u ˆ(ξ), can be extended analytically into the μ-neighborhood of Rn ⊂ Cn , which we denote by Ωμ (Rn ) = {ξ ∈ Cn ; | Im ξ| < μ} ⊂ Cn , and there is Cμ > 0 such that

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N. Boussaïd, A. Comech / Journal of Functional Analysis ••• (••••) ••••••

51

ˆ uL2 (Ωμ (Rn )) ≤ Cμ uL2,μ (Rn ) ,

(A.5)

where Ωμ (Rn ) is interpreted as a region in R2n ∼ = Cn . ˆ on Ωμ as the Fourier–Laplace transform of u Proof. To prove this lemma, one defines u and then computes the L2 -norm for a fixed value of Im ζ. 2 By Lemma A.4, the functions U (ξ) = u (μ (ξ) and V (ξ) = v( μ (ξ) could be extended n n analytically in ξ ∈ R into the strip ξ ∈ C , | 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

(δ)

I2 (ζ) = γa

×Sn−1

dΩθ V (θλ)U (θλ) n−1 λ dλ , 2 2 λ −ζ (2π)n

(A.6)

with γa as on Fig. 2. Clearly, (A.6) is analytic for Re ζ > 0 and Im ζ > 0 (since Im λ2 ≤ 0 while Im ζ 2 > 0). Let us argue that (A.6) can also be extended analytically into the box Kaδ . For λ ∈ γa and ζ ∈ Kaδ , taking 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)| ⎡ ≤

2 ⎢ ⎣ aδ

(A.7)

|V (θλ)|2 |λ|n−1 |dλ|

γa ×Sn−1

dΩθ (2π)n

⎤ 12

|U (θλ)|2 |λ|n−1 |dλ|

dΩθ ⎥ ⎦ , (2π)n

γa ×Sn−1

where the integration in λ over the contour γa could be parametrized by λ(t) = t + iga (t),

t ∈ (a − 2δ, a + 2δ),

dλ = (1 + iga (t)) dt,

(A.8)

so that |dλ| is understood as (1 + (ga (t))2 )1/2 dt. Our assumption that a ≥ 3δ allows us 2 to bound the first factor in (A.7) by aδ ≤ 3δ22 . Moreover, if |ζ| ≥ 2(μ + δ), the first factor in (A.7) is also bounded by 2 2 4 2 < < < , aδ (| Re ζ| − δ)δ (|ζ| − μ − δ)δ |ζ|δ

∀ζ ∈ Kaδ \ D2(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

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52

G(η) =

η ga (|η|)ρ(|η|/δ), |η|

η ∈ Rn ,

(A.9)

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 , We have:

|U (θλ)| |λ| 2

n−1

η ∈ Rn ;

  4μ 2 n2 dΩθ |dλ| ≤ 1 + (2π)n δ

γa ×Sn−1

||η| − a| ≤ 2δ.

|U (η + iG(η))|2 dn η, ||η|−a|<2δ

where we took into account that both |λ/ Re λ| and |dλ/ Re dλ| (which are understood with the aid of the parametrization (A.8) as (t + g(t)2 )1/2 /t and (1 + (ga (t))2 )1/2 , respectively) are bounded by  1 + (g0 )2 ≤ 1 + (4μ/δ)2 . One has: U (η + iG(η)) = Ag u(η), where

Ag u(η) :=

e−ix·η ex·G(η) e−μx u(x) dn x

Rn

is an oscillatory integral operator with the non-degenerate phase function φ(x, η) = x · η and a bounded smooth symbol a(x, η) = ex·G(η)−μx . Note that, by (A.9), x · G(η) − μ x ≤ |x||ga (|η|)| − μ x < 0,

∀(x, η) ∈ Rn × Rn .

By the van der Corput-type arguments applied to Ag A∗g [95, 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

|U (θλ)|2 |λ|n−1 |dλ|

dΩθ ≤ c(μ, δ)u2 . (2π)n

γa ×Sn−1 (δ)

There is a similar bound for V . Thus, there is C = C(μ, δ) > 0 such that |I2 (ζ)| ≤ C(μ,δ) |ζ|δ vu, which is the desired bound.

The estimates on ∂ζj Fμ0 (ζ), j ∈ N, are proved similarly, writing out the derivatives of (ξ 2 − ζ 2 )−1 and proceeding with the same decomposition as in (A.4); the only difference is the contribution from higher powers of ξ 2 − ζ 2 in the denominator.

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N. Boussaïd, A. Comech / Journal of Functional Analysis ••• (••••) ••••••

53

This settles the first part of Proposition A.2. Before we prove the second part of Proposition A.2, we need the following technical lemma. Lemma A.5. Let ρ > 0 and let N ∈ N be odd and satisfy N ≥ 3. The analytic function

ρ

λN −1 dλ , λ2 − ζ 2

FN,ρ (ζ) = 0

ζ ∈ C,

Im ζ > 0,

extends analytically into an open disc Dρ . Moreover, one has ρN −2 |FN,ρ (ζ)| ≤ 2

$

ρ + |ζ| 2 + ln N + π + ln ρ − |ζ|

% ζ ∈ Dρ .

,

(A.10)

2

2

Proof. Using the identity λ2λ−ζ 2 = 1 + λ2ζ−ζ 2 (note that the denominator is nonzero since λ ≥ 0 and Im ζ > 0) and remembering that N is odd, we have:

ρ FN,ρ (ζ) = 0

λN −1 dλ = λ2 − ζ 2

ρ $ λ

N −3

2 N −5

+ζ λ

0

+ ··· + ζ

N −3

ζ N −1 + 2 λ − ζ2

% dλ

 $ %  ζ 2 ρN −4 ρN −2 ζ N −2 ρ−ζ + + · · · + ζ N −3 ρ + Ln + πi . = N −2 N −4 2 ρ+ζ

(A.11)

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, lim Ln

λ→0+

% $ λ−ζ 2λ = lim Ln −1 + = Ln(−1 − 0i) = −πi. λ→0+ λ+ζ 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 ∈ Dρ . The bound (A.10) immediately follows from the inequalities |ζ| < ρ,

1+

N −2 1 1  1 1 1 1 + + ··· + ≤1+ ≤ 1 + ln(N − 2) 3 5 N −2 2 j=2 j 2

(with the understanding that the summation gives no contribution when N = 3), and the bound   $ %   Ln ρ − ζ + πi ≤ π + ln ρ + |ζ|   ρ+ζ ρ − |ζ| valid for ζ ∈ Dρ ∩ C+ (when arg ρ−ζ ρ+ζ ∈ (−π, 0)).

2

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N. Boussaïd, A. Comech / Journal of Functional Analysis ••• (••••) ••••••

Remark A.6. 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. % $ ρ2 1 λ dλ , ln 1 − = λ2 − ζ 2 2 ζ2

ρ 0

ρ 0

λ3 dλ = λ2 − ζ 2

ρ $

ζ 2λ λ+ 2 λ − ζ2

0

%

$ % ζ2 ρ2 ρ2 + ln 1 − 2 , dλ = 2 2 ζ

  which behave like ln − ρζ and ζ 2 ln − ρζ when |ζ|  ρ (hence have a branching point at ζ = 0). 2 Now let us prove the second part of Proposition A.2; 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 Dμ ⊂ C. We pick ρ ∈ (0, μ) and break the integral (A.2) into two parts:

I(ζ) = Rn

V (ξ)U (ξ) n (ρ) (ρ) d ξ = I1 (ζ) + I2 (ζ), ξ2 − ζ 2

(A.12)

where

(ρ)

I1 (ζ) := |ξ|≤ρ (ρ) I2 (ζ)



:= |ξ|>ρ

V (ξ)U (ξ) n d ξ, ξ2 − ζ 2

(A.13)

V (ξ)U (ξ) n d ξ. ξ2 − ζ 2

(A.14)

(ρ)

The function I2 (ζ) in (A.14) is analytic in the disc ζ ∈ Dρ , and moreover for any r ∈ (0, ρ) one has   

  1 V (ξ)U (ξ) n   (ρ) sup |I2 (ζ)| ≤ sup  d ξ ≤ 2 vL2 uL2 . 2 2  ρ − r2 ξ −ζ ζ∈Dr ζ∈Dr   |ξ|>ρ  (ρ)

Let us consider I1 (ζ) defined in (A.13). Since both V (ξ) and U (ξ) are analytic for ξ ∈ Cn , |ξ| < μ, we have the power series expansions V (ξ) =

 α∈Nn 0

Vα ξ α ,

U (ξ) =

 α∈Nn 0

Uα ξ α ,

V (ξ)U (ξ) =



Cα ξ α ,

α∈Nn 0

which are absolutely convergent for |ξ| < μ. Denote λ = |ξ|, θ = ξ/|ξ| ∈ Sn−1 . Then

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N. Boussaïd, A. Comech / Journal of Functional Analysis ••• (••••) ••••••

(ρ) I1 (ζ)

= |ξ|≤ρ



ρ |α|+n−1  V (ξ)U (ξ) n λ dλ α d ξ= Cα θ dΩθ , 2 2 2 2 ξ −ζ λ −ζ n α∈N0

Sn−1

55

(A.15)

0

where θ α = θ1α1 . . . θnαn , θ = (θ1 , . . . , θn ) ∈ Sn−1 . We note that, by parity considerations, the terms corresponding to at least one αj being odd are equal to zero, hence the summation in the right-hand side is only over α ∈ (2N0 )n . We claim that the series (A.15) defines an analytic function in Dρ , and moreover for each r ∈ (0, ρ) there is C > 0 such that (ρ)

sup |I1 (ζ)| ≤ CvL2 (Rn ) uL2 (Rn ) .

ζ∈Dr

We have:



(ρ)

I1 (ζ) =

α∈(2N0 )n

Sn−1





=

θ α dΩθ F|α|+n,ρ (ζ)

Cα

θ α dΩθ R|α|

Cα

α∈(2N0 )n

Sn−1

F|α|+n,ρ (ζ) , R|α|

(A.16)

where R ∈ (ρ, μ) and the analytic function FN,ρ (ζ) was defined in Lemma A.5. By that lemma,   $ % n+|α|−2  (|α| + n)F|α|+n,ρ (ζ)  ρ + |ζ|   ≤ (|α| + n) ρ 2 + ln(|α| + n) + π + ln   ρ − |ζ| R|α| 2R|α| are analytic functions of ζ ∈ Dr , r ∈ (0, ρ), which are bounded uniformly in α ∈ Nn0 and ζ ∈ Dr , by some cr,ρ,R > 0, 0 < r < ρ < R < μ. Using this bound in (A.16), one has: (ρ) |I1 (ζ)|

≤ cr,ρ,R



 α∈(2N0

|Cα θ α R|α| | dΩθ ,

ζ ∈ Dr .

(A.17)

)n

Sn−1

Now we can argue that the series (A.16) is absolutely convergent. To bound the righthand side in (A.17), we use the following lemma with ξ = Rθ. Lemma A.7. For any 0 < R < μ there is CR,μ > 0 such that for any analytic function ) 2n 2n a(ξ) = α∈Nn0 aα ξ α , ξ ∈ Dnμ ⊂ Cn , which has finite norm in L1 (B2n μ ), where Bμ ⊂ R is identified with Dnμ ⊂ Cn , one has sup ξ∈Dn R

 α∈Nn 0

|aα ξ α | ≤ CR,μ aL1 (B2n . μ )

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56

Proof. One can prove Lemma A.7 using the Cauchy estimates

|aα | ≤

dϕn |a(M eiϕ1 , . . . , M eiϕn )| dϕ1 ... , 2π 2π M |α|

S1M ×···×S1M

with M ∈ (R, μ), and changing each of the integrations from ϕj over S1M to the integration over a thin closed annulus included in the region R < |zj | < μ. 2 This lemma, together with the estimate (A.5) from Lemma A.4, shows that, for ζ ∈ Dr , (A.17) is bounded by (ρ)

|I1 (ζ)| ≤ cr,ρ,R vol (Sn−1 ) sup



θ∈Sn−1 α∈(2N )n 0

|Cα θ α R|α| |

≤ cr,ρ,R CR,μ vol (Sn−1 )V U L1 (B2n μ ) ≤ cr,ρ,R CR,μ vol (Sn−1 )V L2 (B2n U L2 (B2n μ ) μ ) ≤ cr,ρ,R CR,μ Cμ2 vol (Sn−1 )vL2,μ (Rn ) uL2,μ (Rn ) , where V (ξ) and U (ξ), ξ ∈ Ωμ (Rn ) ⊂ Cn , denote the analytic continuations of vˆ(ξ) and u ˆ(ξ), ξ ∈ Rn , into the μ-neighborhood of Rn in Cn . We conclude that the series (A.15) is absolutely convergent and therefore defines an analytic function. (ρ) Thus, I1 (ζ) (and hence I(ζ) in (A.12)) has an analytic continuation into the disc (ρ) Dρ for arbitrary ρ ∈ (0, μ), and for any r ∈ (0, ρ) the function I1 (ζ) (and hence I(ζ)) is bounded by C(r)vu as long as ζ ∈ Dr . This concludes the proof of Proposition A.2. 2 Appendix B. Spectrum of the linearized nonlinear Schrödinger 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 [68]. Let us give the essence of the linear stability analysis on the example of the (generalized) nonlinear Schrödinger equation, i∂t ψ = −

1 Δψ − f (|ψ|2 )ψ, 2m

ψ(t, x) ∈ C,

x ∈ Rn ,

n ≥ 1,

t ∈ R,

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 ∈ H 1 (Rn ): φ(x) satisfies the stationary 1 equation ωφ = − 2m Δφ −f (φ2 )φ, 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:

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∂t ρ =

 1 1 − Δρ − ωρ − f (φ2 )ρ − 2f  (φ2 )φ2 Re ρ . i 2m

57

(B.1)

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 Re ρ(t, x) it in the C-linear form, considering its action onto ρ(t, x) = : Im ρ(t, x) 

0 ∂t ρ = −l+

 l− ρ, 0



 Re ρ(t, x) ρ(t, x) = , Im ρ(t, x)

with l− = −

1 Δ − ω − f (φ2 ), 2m

l+ = l− − 2φ2 f  (φ2 ).

(B.2)

If φ ∈ S (Rn ), then by Weyl’s theorem on the essential spectrum one has σess (l− ) = σess (l+ ) = [|ω|, +∞).  Lemma B.2. σ

0 −l+

l− 0



⊂ R ∪ iR.



  2 0 l− l l 0 . Since l− is positive-definite (φ ∈ =− − + 0 l + l− −l+ 0 ker(l− ), being nowhere zero, corresponds to the smallest eigenvalue), we can define the self-adjoint square root of l− ; then

Proof. We consider

 σd

0 −l+

l− 0

2

1/2

1/2

\{0} = σd (l− l+ )\{0} = σd (l+ l− )\{0} = σd (l− l+ l− )\{0} ⊂ R, 1/2

1/2

with the inclusion  due to l− l+ l− being self-adjoint. Thus, any eigenvalue λ ∈  0 l− σd satisfies λ2 ∈ R. 2 −l+ 0 Given the family of solitary waves, φω (x)e−iωt , ω ∈ O ⊂ R, we would like to know at which ω the eigenvalues of the linearized equation with Re λ > 0 appear. Since λ2 ∈ R, such eigenvalues can only be located on the real axis, having  from λ = 0. One  bifurcated 0 l− , with can check that λ = 0 belongs to the discrete spectrum of −l+ 0 

0 −l+

l− 0



 0 = 0, φω



0 −l+

l− 0



   −∂ω φω 0 , = 0 φω

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for all ω which correspond to solitary waves. Thus, if we will restrict our attention to functionswhich are spherically symmetric in x, the dimension of the generalized null 0 l− space of is at least two. Hence, the bifurcation follows the jump in the di−l+ 0 at a particular value of ω mension of its generalized null  space. Such  a jump happens  0 l− ∂ω φω if one can solve the equation α= . This leads to the condition that −l+ 0 0     ∂ω φω 0 l− is orthogonal to the null space of the adjoint to , which contains 0 −l+ 0   φω ; this results in φω , ∂ω φω = ∂ω φω 2L2 /2 = 0. A slightly more careful the vector 0 analysis [25] based on construction 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 ∂ω φω 2L2 becomes positive, leading to a linear instability of the corresponding solitary wave. The opposite condition, ∂ω φω 2L2 < 0, is the Kolokolov stability criterion which guarantees the absence of nonzero real eigenvalues for the nonlinear Schrödinger equation. It appeared in [16,21,55,68,88,105] 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 Δψ − |ψ|2κ ψ, iψ˙ = − 2m

ψ(t, x) ∈ C,

x ∈ Rn . 

 0 l− We need the detailed knowledge of the spectrum of the linearization operator −l+ 0 corresponding to the linearization the solitary wave uκ (x)e−iωt , with uκ a strictly positive 1 spherically symmetric solution to (2.2) and ω = − 2m . Lemma B.3. The dimension of the null space and the generalized null space of the lin 0 l− earization operator from (2.17) is given by −l+ 0  ker

0 −l+

l− 0



 = n + 1,

L

0 −l+

l− 0

 =

 2n + 2,

κ = 2/n;

2n + 4,

κ = 2/n.

Proof. Such computations have appeared in many articles. The relation (2.2) shows that l− uκ = 0. Taking the derivatives of this relation with respect to xi , one also gets l+ ∂xi uκ = 0, 1 ≤ i ≤ n. From [73] or [22, Lemma 2.1] we have that   l+ 0 l+ 0 . dim ker = n +1, hence there are no other vectors in the kernel of 0 l− 0 l− Now let us study the generalized eigenvectors. The relation l− uκ = 0 leads to l− (xi uκ ) = −

1 ∂ i uκ , m x

1 ≤ i ≤ n.

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This shows that    0 l− ∂xi uκ = 0, −l+ 0 0



0 −l+

l− 0



   1 ∂xi uκ 0 = − , 0 x i uκ m

59

1 ≤ i ≤ n. (B.3)

We can not extend this sequence: there is no v such that 

0 −l+

l− 0

    0 v , = i 0 x uκ

since xi uκ is not orthogonal to the kernel of l+ . Indeed, as follows from the identity xi uκ , ∂xi uκ = (−uκ − xi ∂xi uκ ), uκ

(B.4)

(no summation in i), one has xi uκ , ∂xi uκ = − 12 uκ , uκ < 0. By (2.2), the function uκ,λ (x) = λ1/κ uκ (λx) satisfies the identity −

λ2 1 uκ,λ = − Δuκ,λ − u1+2κ κ,λ . 2m 2m

Differentiating this identity with respect to λ at λ = 1 yields 0 = l+ (∂λ |λ=1 uκ,λ ) +

1 1 1 uκ = l+ uκ + x · ∇uκ + uκ . m κ m

(B.5)

This shows that 

0 −l+

l− 0



 0 = 0, uκ



0 −l+

l− 0



   −θ 0 , = 0 uκ

(B.6)

with θ=−

m uκ − mx · ∇uκ , κ

l+ θ = uκ .

(B.7)



 0 l− The relations (B.3) and (B.6) show that dim L ≥ 2n + 2. The dimension −l+ 0 jumps above 2n + 2 in the case when one can find α such that  l− α = θ,

hence

0 −l+

l− 0

    0 θ = . α 0

(B.8)

This happens when θ in (B.7) is orthogonal to ker(l− ) = Span{uκ }. Using the identity (B.4), we see that  1  n 1 1 θ, uκ = − uκ − x · ∇uκ , uκ = − uκ , uκ + uκ , uκ . m κ κ 2

(B.9)

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The right-hand side of (B.9) vanishes when κ = 2/n (that is, when the nonlinear Schrödinger equation is charge-critical). We may choose α to be spherically symmet  ric so that it is orthogonal to ker(l+ ) = Span ∂xi uκ ; 1 ≤ i ≤ n (as the matter 2 of fact, one can take α(x) = − m 2 x uκ (x)); then, by the Fredholm alternative, there is β ∈ L2 (Rn ) such that  l+ β = α,

hence

0 −l+

l− 0



   −β 0 = . 0 α

(B.10)

(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, uκ = β, l+ θ = β, l+ l− α = α, l− α > 0. 2 We also need the following technical result. Lemma B.4. For z ∈ ρ(l− ), the operator (l− − z)−1 : L2 (Rn ) → H 2 (Rn ) extends to a continuous mapping (l− − z)−1 : H −1 (Rn ) → H 1 (Rn ). ∞ Proof. Set a = supx∈Rn uκ (x)2κ . Then there is C < ∞ such that for any ϕ ∈ Ccomp (Rn )

  1 1  1 Δ+ ϕ2H 1 , Cϕ2H 1 ≥ | ϕ, (l− +a)ϕ | ≥ ϕ, − ϕ = 2m 2m 2m

∞ ∀ϕ ∈ Ccomp (Rn ),

hence the self-adjoint operator l− + a : H 2 (Rn ) → L2 (Rn )

(B.11)

is positive-definite and invertible. We can extract its square root, which is a positivedefinite bounded invertible operator (l− + a)1/2 : H 1 (Rn ) → L2 (Rn ); then (B.11) also defines a bounded invertible operator (l− + a)1/2 : H 2 (Rn ) → H 1 (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 : H 1 (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 H 1 (Rn ) → H −1 (Rn ). 2

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Appendix C. The limiting absorption principle for l− We will need the estimates on the resolvent of l− = spaces with exponential weights.

1 2m

Δ − 2m − u2κ κ from (2.18) in the

Lemma C.1. Let Ω ⊂ C be a compact set. Assume that Ω ∩ σd (l− ) = ∅. If z = 1/(2m) is either an eigenvalue or a virtual level of l− , assume further that 1/(2m) ∈ / Ω. Then there is C = C(Ω) > 0 such that the map uκκ ◦ (l− − z)−1 ◦ uκκ : L2 (Rn ) → H 2 (Rn ),

z ∈ C \ σ(l− )

satisfies uκκ ◦ (l− − z)−1 ◦ uκκ L2 →H 2 ≤ C,

∀z ∈ Ω \ [1/(2m), +∞).

Remark C.2. The L2s → L2−s estimate on the resolvent improves as the spectral parameter 2 goes to infinity, while L2s → H−s estimate does not necessarily so (see e.g. Lemma A.1 with different values of ν); this is 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 [1, 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 [58] for n = 3 and from [106, Proposition 7.4.6] for n ≥ 3. The case n ≤ 2, which is left to prove, follows from [59]. We recall the terminology from that article. Given the operators H0 = −Δ and H = H0 + V in L2 (Rn ), we denote

U=

 1, −1,

V ≥ 0; V <0

;

v = |V |1/2 ,

w = U v,

M (ς) = U + v(H0 + ς 2 )−1 v : L2 (Rn ) → L2 (Rn ),

Re ς > 0.

There is the identity (1 − w(H + ς 2 )−1 v)(1 + w(H0 + ς 2 )−1 v) = 1; hence, if M (ς) is invertible, 1 − w(H + ς 2 )−1 v = (1 + w(H0 + ς 2 )−1 v)−1 = (U + v(H0 + ς 2 )−1 v)−1 U = M (ς)−1 U,

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U − w(H + ς 2 )−1 w = M (ς)−1 ,

w(H + ς 2 )−1 w = U − M (ς)−1

κ (cf. [59, Equation (4.8)]). In the case at hand, V = −u2κ κ , U = −1, v = −w = uκ (understood as operators of multiplication); thus,

M (ς) = −1 + uκκ ◦



−

−1 Δ + ς2 ◦ uκκ , 2m

(C.1)

and, when M (ς) is invertible, 2 −1 uκκ ◦ (l− − z)−1 ◦ uκκ = uκκ ◦ (−Δ − u2κ ◦ uκκ = −1 − M (ς)−1 , κ +ς )

(C.2)

with z and ς related by 1 − z = ς 2. 2m Using the expression for the integral kernels of (H0 + ς 2 )−1 in dimensions n ≤ 2 (see [59, Section 3, (3.13) and (3.14)]), the operator M (ς) extends by continuity to the region {ς ∈ C \ {0} ; Re ς ≥ 0}, 1 with a singularity − 2π log(ς) at ς = 0. If −Δ − u2κ κ has no eigenvalue or virtual level at λ = 0, then 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 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. 2

Appendix 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 [2, Theorem IV.1] and [59, Lemma 2.3]). Lemma D.1. Let X1 and X2 be Banach spaces and let A be a closed linear operator on X1 ⊕ X2 with dense domain D(A) ⊂ X1 ⊕ X2 , such that  A=

A11 A21

 A12 , A22

where Aji : Xi → Xj , 1 ≤ i, j ≤ 2 are closed linear operators with domains D(Aj1 ) = {x1 ∈ X1 ; (x1 , 0) ∈ D(A)},

D(Aj2 ) = {x2 ∈ X2 ; (0, x2 ) ∈ D(A)}

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which are dense in X1 and X2 , respectively. Assume that there are Banach spaces W i ⊂ X i ⊂ Yi ,

1 ≤ i ≤ 2,

with continuous dense embeddings, such that Aji extends to a bounded operator from Wi to Yj , with 1 ≤ i, j ≤ 2. Assume further that A11 : W1 → Y1 has a bounded inverse. Let S be the Schur complement of A11 defined by S = A22 − A21 A−1 11 A12 : W2 → Y2 .

(D.1)

Then A (considered as an operator from D(A) to X1 ⊕ X2 ) has a bounded inverse if and only if S : W2 → Y2 is invertible (not necessarily with the bounded inverse) and the −1 −1 −1 restrictions S −1 |X2 , A−1 |X2 , S −1 A21 A−1 A21 A−1 11 A12 S 11 |X1 , and A11 A12 S 11 |X1 define the bounded operators S −1 : X2 → X2 , −1 A−1 : X2 → X1 , 11 A12 S

S −1 A21 A−1 11 : X1 → X2 , −1 A−1 A21 A−1 11 A12 S 11 : X1 → X1 .

Proof. We consider A as a bounded operator from W1 ⊕ W2 to Y1 ⊕ Y2 . Since A11 : W1 → Y1 has a bounded inverse, the operators 

A−1 11 A12 I

I 0



 and

I A21 A−1 11

0 I



are bounded as linear operators in W1 ⊕ W2 and in Y1 ⊕ Y2 , respectively, with bounded inverses given by 

I 0

−A−1 11 A12 I



 and

I

−A21 A−1 11

 0 . I

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 W1 ⊕ W2 to Y1 ⊕ Y2 : 

A11 A= A21 

I

  I A12 = A22 A21 A−1 11

−A21 A−1 11

0 I



A11 A21

A12 A22

0 I





I 0

A11 0

0 S



I 0

 A−1 11 A12 , I

  −A−1 11 A12 = A11 0 I

Explicitly, the inverse of A in terms of the inverse of S is given by

 0 . S

(D.2)

(D.3)

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" A

−1

=

−1 −1 A21 A−1 A−1 11 + A11 A12 S 11

−1 −A−1 11 A12 S

−S −1 A21 A−1 11

S −1

# ;

(D.4)

this expression is known as the Banachiewicz inversion formula. One can see that A (considered as a map from D(A) to X1 ⊕ X2 ) has a bounded inverse if and only if −1 −1 −1 the mappings S −1 , A−1 , S −1 A21 A−1 A21 A−1 11 A12 S 11 , and A11 A12 S 11 are bounded in appropriate spaces. 2 Remark D.2. Swapping the indices in Lemma D.1, we have the following result. Assume that A22 : W2 → Y2 has a bounded inverse. Let T be the Schur complement of A22 defined by T = A11 − A12 A−1 22 A21 : W1 → Y1 .

(D.5)

Then A (considered as an operator from D(A) to X1 ⊕ X2 ) has a bounded inverse if −1 and only if T : W1 → Y1 is invertible and the restrictions T −1 |X1 , A−1 |X1 , 22 A21 T −1 −1 −1 −1 −1 T A12 A22 |X2 , and A22 A21 T A12 A22 |X2 define the bounded operators T −1 : X1 → X1 , −1 A−1 : X1 → X 2 , 22 A21 T

T −1 A12 A−1 22 : X2 → X1 , −1 A−1 A12 A−1 22 A21 T 22 : X2 → X2 .

Similarly to (D.4), the inverse of A in terms of the inverse of T is given in the explicit form by " A

−1

=

T −1

−T −1 A12 A−1 22

−1 −A−1 22 A21 T

−1 −1 A−1 A12 A−1 22 + A22 A21 T 22

# .

(D.6)

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