Amaury L Grimand

published or not. The documents may come

L'évaluation des compétences: paradoxes et faux-semblants d'une instrumentation


Vincent Duchêne

Iva Vukićević

Michael I Weinstein

George C Papanicolaou

Homogenized description of defect modes in periodic structures with localized defects

A spatially localized initial condition for an energy-conserving wave equation with periodic coefficients disperses (spatially spreads) and decays in amplitude as time advances. This dispersion is associated with the continuous spectrum of the underlying differential operator and the absence of discrete eigenvalues. The introduction of spatially localized perturbations in a periodic medium, leads to defect modes, states in which energy remains trapped and spatially localized. In this paper we study weak, O(λ), 0 < λ 1, localized perturbations of one-dimensional periodic Schrödinger operators. Such perturbations give rise to such defect modes, and are associated with the emergence of discrete eigenvalues from the continuous spectrum. Since these isolated eigenvalues are located near a spectral band edge, there is strong scale-separation between the medium period (∼ order 1) and the localization length of the defect mode (∼ order |defect eigenvalue| -1 2 = λ -1 1). Bound states therefore have a multi-scale structure: a "carrier Bloch wave" × a "wave envelope", which is governed by a homogenized Schrödinger operator with associated effective mass, depending on the spectral band edge which is the site of the bifurcation. Our analysis is based on a reformulation of the eigenvalue problem in Bloch quasi-momentum space, using the Gelfand-Bloch transform and a Lyapunov-Schmidt reduction to a closed equation for the near-band-edge frequency components of the bound state. A rescaling of the latter equation yields the homogenized effective equation for the wave envelope, and approximations to bifurcating eigenvalues and eigenfunctions.

Introduction

A spatially localized initial condition for an energy-conserving wave equation with periodic coefficients disperses (spatially spreads) and decays in amplitude as time advances. This (Floquet-Bloch) dispersion is associated with the continuous spectrum (extended states) of the underlying differential operator and the absence of discrete eigenvalues (localized bound states) [START_REF] Kuchment | The mathematics of photonic crystals[END_REF][START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]. The introduction of localized perturbations in a periodic medium leads to defect modes, states in which energy remains trapped and spatially localized. This phenomenon is of great importance in fundamental and applied science -from the existence of stable states of matter in atomic systems to the engineering of materials with desirable energy transport properties through localized doping of ordered materials.

The process by which the system undergoes a transition from one with only propagating delocalized states to one which supports both localized and propagating states is associated with the 1 arXiv:1301.0837v2 [math-ph] 28 Jul 2014 emergence or bifurcation of discrete eigenvalues from the continuous spectrum associated with the unperturbed periodic structure. In this paper, we discuss this bifurcation phenomenon in detail for the Schrödinger operator

HQ = -∂ 2 x + Q(x) (1.1)
where Q(x) is a continuous, real-valued, periodic potential:

Q(x + 1) = Q(x).
(1.

2)

The spectrum, spec(HQ), of the Schrödinger operator is continuous and is the union of closed intervals called spectral bands [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]. The complement of the spectrum is a union of open intervals called spectral gaps. The spectrum is determined by the family of self-adjoint eigenvalue problems parameterized by the quasi-momentum k ∈ (-1/2, 1/2]:

HQu(x; k) = E u(x; k) , (1.3)
u(x + 1; k) = e 2πik u(x; k) .

(1.4)

That is, we seek k-pseudo-periodic solutions of the eigenvalue equation. For each k ∈ (-1/2, 1/2], the self-adjoint eigenvalue problem (1.3)-(1.4) has discrete eigenvalue-spectrum (listed with multiplicity):

E0(k) ≤ E1(k) ≤ • • • ≤ E b (k) ≤ . . . (1.5)
with corresponding k-pseudo-periodic eigenfunctions:

u b (x; k) = e 2πikx p b (x; k), p b (x + 1; k) = p b (x; k), b ≥ 0. (1.6)
The b th spectral band is given by

B b = k∈(-1/2,1/2] E b (k). (1.7)
The spectrum of HQ is given by: spec(HQ) = Since the boundary condition (1.4) is invariant with respect to k → k + 1, the functions E b (k) can be extended to all R as periodic functions of k. The minima and maxima of E b (k) occur at k = k * ∈ {0, 1/2}; see Figure 1. In cases where extrema border spectral gap, we have that ∂ 2 k E b (k * ) is either strictly positive or strictly negative [START_REF] Eastham | The spectral theory of periodic differential equations[END_REF][START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]; see Lemma 2.2. Consider now the perturbed operator HQ+V , where V (x) is sufficiently localized in space. By Weyl's theorem on the stability of the essential spectrum, one has spec cont (HQ+V ) = spec cont (HQ); see [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]. The effect of a localized perturbation is to possibly introduce discrete eigenvalues into the open spectral gaps. Note that in our setting, HQ+V does not have discrete eigenvalues embedded in its continuous spectrum; see [START_REF] Rofe-Beketov | A test for finiteness of the number of discrete levels introduced into the gaps of a continuous spectrum by perturbation of a periodic potential[END_REF], [START_REF] Gesztesy | A short proof of Zheludev's theorem[END_REF].

In this paper we present a detailed study of the bifurcation of localized bound states into gaps of the continuous spectrum induced by a small and localized perturbation of HQ:

H Q+λV ≡ -∂ 2 x + Q(x) + λV (x) , λ > 0, (1.9)
where λ is taken sufficiently small. Here Q(x) is a continuous, 1-periodic function defined on R and V (x) is spatially localized. We next turn to a summary of our results. See Theorem 3.1 and Theorem 3.4 for detailed statements.

Let E * = E b * (k * ), k * ∈ {0, 1/2}, denote an endpoint (uppermost or lowermost) of the (b * ) th spectral band, bordering a spectral gap. We show that under the condition:

∂ 2 k E b * (k * ) × ˆR |u b * (x; k * )| 2 V (x) dx < 0 , (1.10)
the following holds: There exists a positive number, λ0, such that for all 0 < λ < λ0, H Q+λV has a simple discrete eigenvalue For 0 < λ < λ0, ψ λ (x), the eigenstate corresponding to the eigenvalue, E(λ), is well-approximated in L ∞ by, g0(λx), where g0(y) denotes the unique eigenstate of the homogenized operator Remark 1.1. The notion of effective mass is well known in condensed matter physics [START_REF] Ashcroft | Solid State Physics[END_REF]. The effective mass for an evolving wave-packet may be derived by multiscale perturbation theory and is related the general problem of homogenization of periodic structures; see the very influential book of Bensoussan, Lions & Papanicolaou [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF]; see also [START_REF] Birman | Two-dimensional periodic Pauli operator. The effective masses at the lower edge of the spectrum[END_REF][START_REF] Birman | On the homogenization procedure for periodic operators in a neighborhood of an edge of an internal gap[END_REF][START_REF] Birman | Second order periodic differential operators. Threshold properties and homogenization[END_REF][START_REF] Allaire | Homogenization of the Schrödinger equation and effective mass theorems[END_REF][START_REF] Birman | Homogenization of a multidimensional periodic elliptic operator in a neighborhood of an edge of an inner gap[END_REF].

E(λ) = E * + λ 2 µ + O(λ 2+α ), α > 0 . ( 1
Remark 1.2. For the case Q ≡ 0, then HQ = H0 = -∂ 2 x and its spectrum consists of a semiinfinite interval, spec(H0) = [0, ∞), the union of touching bands with no finite length gaps. Furthermore, p b (x; k) ≡ 1, for all |k| ≤ 1/2 and b ≥ 0. The only band edge which borders a gap is located at E * = E0(0) = 0, where we have: k * = 0, E0(k) = 4π 2 k 2 and ∂ 2 k E0(k * ) = 8π 2 . In this case, our results describe the bifurcation of an eigenvalue from the edge of the continuous spectrum of H0 induced by a small and localized perturbation: H λV = -∂ 2

x + λV , under the condition ´R V < 0. The homogenized operator is

H 0,eff = - d 2 dy 2 + λ ˆR V dx• ; (1.16)
see the discussion below of [START_REF] Simon | The bound state of weakly coupled Schrödinger operators in one and two dimensions[END_REF]. periodic. The center (resp. righ) panel corresponds spec(H Q+λV ), where λV is small, localized negative (resp. positive).

Previous related work

The case Q ≡ 0, where H Q+λV = -∆ + V (x) was considered by Simon [START_REF] Simon | The bound state of weakly coupled Schrödinger operators in one and two dimensions[END_REF] in one and two spatial dimensions. In one dimension, it is proved that if V is sufficiently localized and -∞ < ´R V < 0, then H λV has a small negative eigenvalue E(λ) of order λ 2 ; see the Corollary 3.3 and the discussion following it. The case of perturbations of one-dimensional periodic Schrödinger operators (Q nontrivial, 1-periodic) is treated by Gesztesy & Simon [START_REF] Gesztesy | A short proof of Zheludev's theorem[END_REF], where sufficient conditions are given for the bifurcation of eigenvalues in the gaps of the continuous spectrum. Borisov and Gadyl'shin [START_REF] Borisov | On the spectrum of a periodic operator with a small localized perturbation[END_REF] obtain results closely related to the current work, although using very different methods. A formal asymptotic study, in terms of a Floquet-Bloch decomposition, in one and two spatial dimensions was given in Wang et. al. [START_REF] Wang | Two-dimensional defect modes in optically induced photonic lattices[END_REF]. Parzygnat et. al. [START_REF] Parzygnat | Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps[END_REF] formulate a variational principle for defect modes with frequencies in spectral gaps. They use formal trial function arguments to show the existence of such defect modes in spatial dimensions one and two. By formal asymptotic arguments, they obtain the condition (1.10), for the case of the first spectral gap. Deift & Hempel [START_REF] Deift | On the existence of eigenvalues of the schrödinger operator h-λw in a gap of σ(h)[END_REF] obtained results on the existence and number of eigenstates in spectral gaps for operators of the general type H -λW , where H has a band spectrum and W is bounded. Figotin & Klein [START_REF] Figotin | Localized classical waves created by defects[END_REF][START_REF] Figotin | Midgap defect modes in dielectric and acoustic media[END_REF] studied localized defect modes in context of acoustic and electromagnetic waves. Results on bound states and scattering resonances of one-dimensional Schrödinger operators with compactly supported potentials appear in work of Bronski & Rapti [START_REF] Bronski | Counting defect modes in periodic eigenvalue problems[END_REF] and Korotyaev [START_REF] Korotyaev | 1d Schrödinger operator with periodic plus compactly supported potentials[END_REF][START_REF] Korotyaev | Resonance theory for perturbed Hill operator[END_REF], respectively. Bifurcations of defect modes into spectral gaps was considered in dimensions d = 1, 2 and 3 by Hoefer & Weinstein [START_REF] Hoefer | Defect modes and homogenization of periodic schrödinger operators[END_REF] for operators of the form

-∆ + Q(x) + ε 2 V (εx)
, where Q is periodic on R d and V is spatially localized. This scaling was motivated by work of Ilan & Weinstein [START_REF] Ilan | Band-edge solitons, nonlinear Schrödinger/Gross-Pitaevskii equations and effective media[END_REF] on the bifurcation of nonlinear bound states from continuous spectra for the nonlinear Schrödinger / Gross-Pitaevskii equation. The works [START_REF] Ilan | Band-edge solitons, nonlinear Schrödinger/Gross-Pitaevskii equations and effective media[END_REF][START_REF] Hoefer | Defect modes and homogenization of periodic schrödinger operators[END_REF] employ the general Lyapunov-Schmidt reduction strategy used in the present work; see also [START_REF] Pelinovsky | Justification of the the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential[END_REF][START_REF] Dohnal | Coupled mode equations and gap solitons for the 2D Gross-Pitaevskii equation with a non-separable periodic potential[END_REF][START_REF] Dohnal | Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential[END_REF].

Outline, remarks on the proof and future directions

In Section 2 we present background material concerning spectral properties of Schrödinger operators with periodic potentials defined on R. In Section 3 we give precise technical statements of our main results: Theorem 3.1 and Theorem 3.4.

Our strategy of proof is to transform the eigenvalue problem, using the appropriate spectral transform (Fourier or Floquet-Bloch), to a formulation in frequency (quasi-momentum) space. Anticipating a bifurcation from the spectral edge, we express the eigenvalue problem in terms of coupled equations governing the frequency components located near the band edge and those which are far from the band edge. The precise frequency cutoff depends on the small parameter, λ. We employ a Lyapunov-Schmidt reduction strategy [START_REF] Nirenberg | Topics in Nonlinear Functional Analysis[END_REF] in which we solve for the far-frequency components as a functional of the near-frequency components. This yields a reduction to a closed bifurcation equation for the near-frequency components. In contrast to classical applications of this strategy, our reduced equation is infinite dimensional. For λ small, in an appropriate scaled limit, the bifurcation equation is asymptotically exactly solvable; it is the eigenvalue problem for the homogenized / effective operator H b * ,eff . In Section 4, we prove a general technical lemma, crucial to the analyses of Sections 5 and 6, covering the kinds of bifurcation equations which arise. Finally, Appendices A and B contain the proof of results stated in Lemmata 2.2 and 2.3, and in Appendix C we give proofs, by a bootstrap method, of Corollary 3.3 and Corollary 3.6 which contain more detailed expansions and sharper error terms for the bifurcating eigenstates than those in Theorem 3.1 and Theorem 3.4 .

We conclude this section with several possible extensions of the present work.

1. The results of this paper describe the bifurcation of eigensolutions in the case where the perturbing potential is small in the strong sense (in norm). What of the case where the perturbing potential converges weakly to zero? This corresponds to the question of the effective behavior of high-contrast microstructures. In [START_REF] Duchêne | Scattering and localization properties of highly oscillatory potentials[END_REF], the authors consider a class of problems, depending on a small parameter, ε, including the case where the potential, q (ε) (x) = q(x, x/ε), converges weakly as ε tends to zero. In particular, we considered the small ε limit of the scattering and time-evolution properties for operators of the form H (ε) = -∂ 2 x + q(x, x/ε), where y → q(•, y) is oscillatory (including periodic and certain almost periodic cases) and x → q(x, •) is spatially localized. An important subtlety arises in the case where qav(x) = ´R q(x, y)dy ≡ 0, i.e. q ε tends to zero weakly; see [START_REF] Duchêne | Scattering, homogenization and interface effects for oscillatory potentials with strong singularities[END_REF] for the case where qav(x) = 0 is generic. In this case, classical homogenization theory breaks down at low energies. Indeed, the homogenized operator, obtained by averaging the potential over its fast variations, is H0 = -∂ 2

x , which does not capture key spectral and scattering information. Among these are the low energy behavior of the transmission coefficient (related to the spectral measure) and the existence of a bifurcating bound state at a very small negative energy. We show that the correct behavior is captured by an effective Hamiltonian with effective potential well:

H (ε) eff = -∂ 2 y -ε 2 Λ eff (y), Λ eff (y) > 0.
Using Theorem 3.1 and the results of [START_REF] Simon | The bound state of weakly coupled Schrödinger operators in one and two dimensions[END_REF], we conclude that H (ε) has a bound state with negative energy of the order ε 4 , with a precise expansion for ε small. Thus, it would be interesting to use our approach in order to extend the results of the present paper to families of potentials, q ε , which converge weakly to a nontrivial periodic potential, Q(x); see [17, in progress].

2. Further, in [START_REF] Duchêne | Scattering and localization properties of highly oscillatory potentials[END_REF] a multi-scale local energy time-decay estimate, for localized initial conditions orthogonal to the bound state, in which the different dispersive time-dynamics on different time-scales is explicit. In particular, the decay rate is O(t -1 2 ) for times t ε -2 and O(t -3 2 ) for t ≥ ε -2 . We believe that our methods can be extended to give detailed properties of the resolvent -∂ 2

x + Q + λV -E -1 and therefore the spectral measure [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF] near the band edges. Such information could be used to derive the detailed dispersive time-decay behavior.

However, the decay estimates of the type obtained in [START_REF] Duchêne | Scattering and localization properties of highly oscillatory potentials[END_REF] can be expected to hold only for initial conditions which are spectrally localized near band edges. Initial conditions with spectral components away from the band edge can sample a regime where, for Q non-zero, the dispersion relation has higher degeneracy, yielding different (slower) dispersive time-decay [START_REF] Firsova | On the time decay of a wave packet in a one-dimensional finite band periodic lattice[END_REF][START_REF] Cai | Dispersion for Schrödinger operators with one gap periodic potentials in R, Dynamics Part[END_REF][START_REF] Cuccagna | Dispersion for Schrödinger equation with periodic potential in 1D[END_REF].

3. Finally, it would be of interest to extend the methods of the current paper to the study of bifurcations of eigenvalues for multiplicatively small or weakly convergent spatially localized perturbations of the higher-dimensional periodic Schrödinger operator, -∆ + Q. In spatial dimension n = 2 and the case Q = 0, Simon [START_REF] Simon | The bound state of weakly coupled Schrödinger operators in one and two dimensions[END_REF] proved that the bound state generated by a multiplicatively small perturbation is exponentially close to the edge of the continuous spectrum. Such results has been extended by Borisov [8] in the periodic (Q nontrivial) case. Formal asymptotics were obtained in Wang et. al. [START_REF] Wang | Two-dimensional defect modes in optically induced photonic lattices[END_REF]. In spatial dimensions n ≥ 3, it is well known that for sufficiently small λ, -∆ + λV does not have discrete spectrum, by Cwikel-Lieb-Rozenblum bound. Finally, Parzygnat et. al. [START_REF] Parzygnat | Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps[END_REF] also treat the case of dimensions n ≥ 3, where the defect potential, V is localized along in one or two dimensions.

Definitions and notation

We denote by C a constant, which does not depend on the small parameter, λ.

χ δ (ξ) = χ (|ξ| < δ) ≡ 1, |ξ| < δ 0, |ξ| ≥ δ , χ δ (ξ) = χ(|ξ| < δ) ≡ 1 -χ(|ξ| < δ) . (1.17)
For f, g ∈ L2 (R), the Fourier transform and its inverse are given by F{f }(ξ) ≡ f (ξ) = ˆR e -2πixξ f (x)dx, F -1 {g}(x) ≡ ǧ(x) = ˆR e 2πixξ g(ξ)dξ.

T and T -1 denote the Gelfand-Bloch transform and its inverse; see Section 2. L p,s (R) is the space of functions F : R → R such that

(1 + | • | 2 ) s/2 F ∈ L p (R), endowed with the norm F L p,s (R) ≡ (1 + | • | 2 ) s/2 F L p (R) < ∞, 1 ≤ p ≤ ∞. (1.18) W k,p (R) is the space of functions F : R → R such that ∂ j x F ∈ L p (R) for 0 ≤ j ≤ k, endowed with the norm F W k,p (R) ≡ k j=0 ∂ j x F L p (R) < ∞, 1 ≤ p ≤ ∞.

Floquet-Bloch states

We seek solutions of the k-pseudo-periodic eigenvalue problem

(-∂ 2 x + Q(x))u(x) = Eu(x) , u(x + 1) = e 2πik u(x) , (2.1)
for k ∈ (-1/2, 1/2], the Brillouin zone. Setting u(x; k) = e 2πikx p(x; k), we equivalently seek eigensolutions (p(x; k), E) of the periodic elliptic boundary value problem:

-(∂x + 2πik) 2 + Q(x) p(x; k) = E(k)p(x; k), p(x + 1; k) = p(x; k) (2.2) for each k ∈ (-1/2,
(x; k) ≡ e 2πikx p b (x; k), b ∈ N, -1/2 < k ≤ 1/2} is complete in L 2 (R), i.e. for any f ∈ L 2 (R), f (x) - 0≤b≤N ˆ1/2 -1/2 u b (•, k), f L 2 (R) u b (x; k) dk → 0 in L 2 (R) as N ↑ ∞.
Recall further that the spectrum of -∂ 2 x + Q(x) is continuous, and equal to the union of the closed intervals, the spectral bands; see (1.7), (1.8).

Definition 2.1. We say there is a spectral gap between the b th and (b + 1) st bands if

sup |k|<1/2 |E b (k)| < inf |k|<1/2 |E b+1 (k)|. (2.3)
Our study of eigenvalue bifurcation from the band edge E * ≡ E b * (k * ) into a spectral gap, requires regularity and detailed properties of E b (k) near its edges. These are summarized in the following results (see a sketch of E b (k) in Figure 1, left panel). Proofs and references to proofs are given in Appendices A and B. Lemma 2.2. Assume E b (k * ) is an endpoint of a spectral band of -∂ 2

x + Q(x), which borders on a spectral gap; see (2.3). Then k * ∈ {0, 1/2} and the following results hold:

1. E b (k * ) is a simple eigenvalue of the eigenvalue problem (2.1).

2.

b even: E b (0) corresponds to the left (lowermost) end point of the band, E b (1/2) corresponds to the right (uppermost) end point. b odd: E b (0) corresponds to the right (uppermost) end point of the band, E b (1/2) corresponds to the left (lowermost) end point.

3. ∂ k E b (k * ) = 0, ∂ 3 k E b (k * ) = 0; 4. b even: ∂ 2 k E b (0) > 0, ∂ 2 k E b (1/2) < 0; b odd: ∂ 2 k E b (0) < 0, ∂ 2 k E b (1/2) > 0; Lemma 2.3. Let E b (k1
) denote a simple eigenvalue; thus k1 = k * as above applies. Then, the mappings k → E b (k), k → u b (x; k), with u b normalized, can be chosen to be analytic for k in a sufficiently small complex neighborhood of k1. Moreover, for k real and in this neighborhood (E b (k), u b (x; k)) are Floquet-Bloch eigenpairs.

The Gelfand-Bloch transform

Given f ∈ L 2 (R), we introduce the Gelfand-Bloch transform T and its inverse as follows

T {f (•)} = f (x; k) = n∈Z e 2πinx f (k + n), T -1 { f (x; •)}(x) = ˆ1/2 -1/2 e 2πixk f (x; k)dk. One can check that T -1 T = Id. Let u(x; k) = e 2πikx p(x; k) (2.4)
denote a Floquet-Bloch mode. Then, by the Poisson summation formula, we have that

u(•, k), f L 2 (R) = p(•, k), f (•; k) L 2 per ([0,1]) .
Define

T b {f }(k) ≡ p b (•; k), f (•; k) L 2 per ([0,1]) ≡ ˆ1 0 p b (x; k) f (x; k)dx. (2.5)
By completeness of the {p b (x; k)} b≥0 , the spectral decomposition of f ∈ L 2 (R) in terms of Floquet-Bloch states is

f (x; k) = b≥0 T b {f }(k)p b (x; k) , f (x) = b≥0 ˆ1/2 -1/2 T b {f }(k)u b (x; k)dk.
Recall the Sobolev space, H s , the space of functions with square integrable derivatives up to order s. It is natural to construct the following X s norm in terms of Floquet-Bloch states:

φ 2 X s ≡ ˆ1/2 -1/2 b≥0 1 + |b| 2 s |T b {φ}(k)| 2 dk. (2.6)
Proposition 2.4. H s (R) is isomorphic to X s for s ≥ 0. Moreover, there exist positive constants C1, C2 such that for all φ ∈ H s (R), we have C1 φ

H s (R) ≤ φ X s ≤ C2 φ H s (R) . Proof. Since E0(0) = inf spec(-∂ 2 x + Q), then L0 ≡ -∂ 2 x + Q -E0 (
0) is a non-negative operator and H s (R) has the equivalent norm defined by φ H s ≈ (I + L0) s/2 φ L 2 . Using orthogonality, it follows that

φ 2 H s ≈ (I + L0) s/2 φ 2 L 2 = b≥0 ˆ1/2 -1/2 |T b {φ}(k)| 2 |1 + E b (k) -E0(0)| s dk ≈ b≥0 1 + |b| 2 s ˆ1/2 -1/2 |T b {φ}(k)| 2 dk ≡ φ 2 X s .
The last line follows from the Weyl asymptotics E b (k) ∼ b 2 ; see, for example, [START_REF] Courant | Methods of Mathematical Physics[END_REF]. This completes the proof of Proposition 2.4 .

We conclude this section with a Lemma, which gives various estimates on the Floquet-Bloch states of HQ and the spectrum of H Q+λV , for a class of periodic potentials, Q, and localized potentials, V . These estimates are used within the proof of Theorem 3.4, in Section 6.

Lemma 2.5. Assume that Q is continuous, 1-periodic, and V is such that

(1 + | • |)V (•) ∈ L 1 .
Let Ω be a small neighborhood of k1 a simple eigenvalue, such that Lemma 2.3 applies. Then one has:

(a) sup k∈(-1 2 , 1 2 ] p b (•; k) L ∞ ≤ sup k∈(-1 2 , 1 2 ] n∈Z p b (•; k), e 2πin• L 2 ([0,1]) < ∞, (2.7a) (b) sup k∈Ω ∂ k p b (•; k) L ∞ ≤ sup k∈Ω n∈Z ∂ k p b (•; k), e 2πin• L 2 ([0,1]) < ∞. ( 2
= n∈Z ˆ1 0 p b (x; k)e -2πinx dx .
Since k ∈ (-1/2, 1/2], we can use integration by parts for n = 0, the Cauchy-Schwarz inequality and equation (2.2) for p b (x; k) to obtain

n∈Z p b (•; k), e 2πin• L 2 ([0,1]) ≤ p b (x; k) L 2 ([0,1]) 1 L 2 ([0,1]) + n∈Z\{0} ˆ1 0 (Q(x) -E b (k))p b (x; k) 1 2πi(n -k) 2 e -2πinx dx ≤ 1 + n∈Z\{0} 1 4π 2 (n -k) 2 (Q(•) -E b (k))p b (•; k) L 2 ([0,1]) . Thus, sup k∈(-1/2,1/2] p b (•; k) L ∞ ≤ sup k∈(-1/2,1/2] n∈Z p b (•; k), e 2πin• L 2 ([0,1]) < ∞.
We now turn to the study of ∂ k p b (x; k) in (b). Differentiating (2.2) with respect to k yields

-(∂x + 2πik) 2 + Q(x) ∂ k p b (x; k) = E b (k)∂ k p b (x; k) + ∂ k E b (k) + 4πi(∂x + 2πik) p b (x; k).
Following the same method as above yields

∂ k p b (•; k) L ∞ ≤ n∈Z ∂ k p b (•; k), e 2πin• L 2 ([0,1]) ≤ C( Q L ∞ , E b (k)) ∂ k p b (x; k) L 2 ([0,1]) + C(∂ k E b (k), Q L ∞ , E b (k)). The finiteness of ∂ k p b (•; k) L 2 ([0,1]) and ∂ k E b (k) for k ∈ Ω is a consequence of Lemma 2.3; thus (b) follows.

Bifurcation of defect states into gaps; main results

Consider the eigenvalue problem:

-∂ 2 x + Q(x) + λV (x) ψ λ = E λ ψ λ , ψ ∈ L 2 (R),
where Q(x) is continuous, 1-periodic, λ > 0 is small, and V (x) is spatially localized. Our first result concerns the case where Q ≡ 0:

Theorem 3.1 (Q ≡ 0). Let V be such that V ∈ W 1,∞ (R); thus ´R(1 + |x|)|V (x)| dx < ∞ suffices. Assume V (0) = ´R V < 0.
There exists positive constants λ0 and C(V, λ0), such that for all 0 < λ < λ0, there exists an eigenpair (E λ , ψ λ ), solution of the eigenvalue problem

-∂ 2 x + λV (x) ψ λ (x) = E λ ψ λ (x) (3.1)
with negative eigenvalue of the order λ 2 . Specifically,

E λ -- λ 2 4 ˆR V 2 ≤ Cλ 5/2 , (3.2)
sup x∈R ψ λ (x) -exp λ 2 ˆR V |x| ≤ Cλ 1/2 . (3.
3)

The eigenvalue, E λ , is unique in the neighborhood defined by (3.2), and the corresponding eigenfunction, ψ, is unique up to a multiplicative constant.

Remark 3.2. Theorem 3.1 shows, and is essentially proved by demonstrating, that for small positive λ, the leading order behavior of the eigenstate E λ , ψ λ (x) is a scaling of the unique eigenstate of the attractive Dirac delta potential:

E λ , ψ λ (x) ≈ λ 2 θ 2 0 , g0 (λx)
,

where θ0 = -1 2 ´R V > 0 and g0(y) = e -θ 0 |y| satisfy -∂ 2 y + ˆR V • δ(y) g0(y) = -θ 2 0 g0(y). (3.4)
The error bounds in Theorem 3.1 are not optimal. However, the bootstrap argument of Appendix C can be used to recover a higher order expansion on E λ , similar to that obtained in [START_REF] Simon | The bound state of weakly coupled Schrödinger operators in one and two dimensions[END_REF].

Corollary 3.3. Assume (1 + |x| 2 )V ∈ L 1 ,
and V (0) = ´R V (z) dz < 0 . Then E λ , as defined in Theorem 3.1, satisfies the precise estimate:

E λ = -λ 2 [θ(λ)] 2 , with θ(λ) = - 1 2 ˆR V - 1 4 λ ¨R2 V (x)|x-y|V (y) dxdy + O(λ 3/2 ) . (3.5)
Simon [START_REF] Simon | The bound state of weakly coupled Schrödinger operators in one and two dimensions[END_REF] and Klaus [START_REF] Klaus | On the bound state of Schrödinger operators in one dimension[END_REF] prove expansion (3.5), under the conditions:

(1 + |x|)|V (x) ∈ L 1 (R)
and ´R V ≤ 0, with the error term o(λ). Corollary 3.3 gives a sharper error term under a more stringent decay condition on V . That Theorem 3.1 implies Corollary 3.3 is proved in Appendix C .

Theorem 3.4 ( Q non-trivial, 1-periodic). Let Q be continuous, 1-periodic, and let V be such that ´R(1 + |x|)V (x)dx < ∞ and V ∈ L ∞ . Let E b * : k ∈ (-1/2, 1/2] → R denote the band dispersion function associated with the (b * ) th band of the continuous spectrum of -∂ 2 x + Q(x). Fix a spectral band edge of the (b * ) th band; thus E * = E b * (k * ), where k * = 0 or k * = 1/2 (see Lemma 2.2).
Assume either

∂ 2 k E b * (k * ) > 0 and ˆR |u b * (x; k * )| 2 V (x)dx < 0, (3.6
)

or ∂ 2 k E b * (k * ) < 0 and ˆR |u b * (x; k * )| 2 V (x)dx > 0. (3.7)
Then, there is a positive constants, λ0 and C = C(λ0, V, Q), such that for all λ < λ0, the following assertions hold:

1. There exists an eigenpair E λ , ψ λ (x) of the eigenvalue problem

-∂ 2 x + Q(x) + λV (x) ψ λ (x) = E λ ψ λ (x), ψ λ ∈ L 2 (R) . (3.8) 2. Define α0 ≡ ´∞ -∞ |u b * (x; k * )| 2 V (x)dx 1 4π 2 ∂ 2 k E b * (k * ) < 0, (3.9)
where the inequality holds by (3.6) and (3.7). Then, E λ and ψ λ (x) satisfy the following approximations:

E λ -E b * (k * ) + λ 2 E2 ≤ Cλ 2+1/4 , (3.10
)

sup x∈R ψ λ (x) -u b * (x; k * ) exp(λα0|x|) ≤ Cλ 1/4 , (3.11)
where

E2 = - ´∞ -∞ |u b * (x; k * )| 2 V (x)dx 2 1 2π 2 ∂ 2 k E b * (k * ) . (3.12)
Note that the direction of bifurcation of E λ is given by:

sgn (E2) = -sgn ∂ 2 k E b * (k * ) .
3. The eigenstate, (E λ , ψ λ ), is unique (up to a multiplicative constant for ψ λ ) in the neighborhood defined by (3.10), (3.11).

Remark 3.5. By Theorem 3.4, the bifurcating eigenvalue E λ lies in the spectral gap of -∂ 2

x +Q(x) at a distance O(λ 2 ) near the spectral edge E * ; see Figure 1. Moreover, E2 is the unique eigenvalue and g0(y) = e α 0 |y| is the unique (up to multiplication by a constant) eigenfunction of the effective (homogenized) Hamiltonian:

H eff = - d dy 1 8π 2 ∂ 2 k E b * (k * ) d dy + ˆ∞ -∞ |u b * (x; k * )| 2 V (x)dx × δ(y).
The following refinement of Theorem 3.4 can be proved via the bootstrap argument presented in Appendix C. Corollary 3.6. Assume ´R(1 + |x| 2 )V (x)dx < ∞ and that the hypotheses of Theorem (3.4) hold.

Then,

E λ -E b * (k * ) = λ 2 (E2 + λE3) + O(λ 3+1/4 ) = -λ 2 8π 2 ∂ 2 k E b * (k * ) [Θ(λ)] 2 , (3.13)
where E2 is as in (3.12),

E3 ≡ -8π 4 (∂ 2 k E b * (k * )) 2 ˆ∞ -∞ |u b * (x; k * )| 2 V (x) dx × ¨R2 V (x)|u b * (x; k * )| 2 |x -y||u b * (y; k * )| 2 V (y) dx dy , and
Θ(λ) = - 1 2 ˆR |u b * (x; k * )| 2 V (x) dx (3.14) - 1 4 λ 8π 2 ∂ 2 k E b * (k * ) ¨R2 V (x)|u b * (x; k * )| 2 |x -y||u b * (y; k * )| 2 V (y) dx dy + O(λ 1+1/4 ).
Remark 3.7. For the case Q ≡ 0, the spectrum consists of only one semi-infinite band which we can label the b = 0 band. In this case, u0(x; k * = 0) = 1 and E0(k) = 4π 2 k 2 . Therefore, to leading order, relation (3.14) simplifies to the result of Corollary 3.3 and the two results are consistent.

Key general technical results

In this section, we study the operator L0[θ], defined by:

f (ξ) → L0[θ] f (ξ) ≡ 4π 2 Aξ 2 + θ 2 f (ξ) -B χ |ξ| < λ -β ˆR χ |η| < λ -β f (η) dη. (4.1)
Here, A, B and β are fixed positive constants. The operator L0[θ] appears in the bifurcation equations we derived via the Lyapunov-Schmidt reduction; see Section 1.2. In x-space, we have that L0[θ] is a rank one perturbation of -A∂ 2 y + θ 2 :

L0[θ]f ≡ (-A∂ 2 y + θ 2 )f (y) - 2B λ β 2 λ β sinc 2π λ β • , f (•) L 2 sinc 2πy λ β , (4.2)
where sinc(z) = sin(z)/z. L0[θ] is a band-limited regularization of the operator:

H A,B + θ 2 f ≡ -A∂ 2 y -Bδ(y) + θ 2 f , (4.3)
appearing in the effective equations governing the leading order behavior of bifurcating eigenstates; see Remarks 3.2 and 3.5.

4.1

The operator L 0 Lemma 4.1. Fix constants A > 0, B > 0 and β > 0. Define, for θ 2 > 0, the linear operator

f (ξ) → L0[θ] f (ξ) ≡ 4π 2 Aξ 2 + θ 2 f (ξ) -B χ |ξ| < λ -β ˆR χ |η| < λ -β f (η) dη. (4.4) Note that L0[θ] : L 1 (R) → L 1,-2 (R); see (1.

18).

1. There exists a unique θ 2 0 > 0 such that L0[θ0] has a non-trivial kernel. 2. The "eigenvalue" θ 2 0 is the unique positive solution of

1 -B ˆR χ |ξ| < λ -β 4π 2 Aξ 2 + θ 2 0 dξ = 0 . (4.5)
3. The kernel of L0[θ0] is given by:

kernel L0[θ0] = span f0(ξ) , where f0(ξ) ≡ χ |ξ| < λ -β 4π 2 Aξ 2 + θ 2 0 . (4.6)
4. θ0 = θ0(λ) can be approximated as follows:

θ0 - B 2 √ A ≤ θ0 2π 2 B A λ β . (4.7) 5. Define g(x) = exp(α0|x|), with α0 = -B 2A < 0. Then one has sup x∈R F -1 f0 (x) - 1 B g(x) ≤ C(A, B)λ β . (4.8)
Proof. First note, by rearranging terms in the equation L0[θ0] g = 0, that any element, g(ξ), of the kernel of L0[θ], is a constant multiple of the function

f λ (ξ; θ) ≡ χ(|ξ| < λ -β ) × (4π 2 Aξ 2 + θ 2 ) -1 .
Thus, if g is non-trivial then it is strictly positive or strictly negative and therefore ´R g = 0. Next, note that a necessary condition for g to lie in the kernel of L0[θ] is that equation (4.5) holds. To see this, divide the equation L0[θ0] g = 0 by 4π 2 Aξ 2 + θ0 2 and integrate dξ over R. This yields:

ˆ∞ -∞ g(ξ) dξ × 1 -B ˆ∞ -∞ χ |ξ| < λ -β 4π 2 Aξ 2 + θ 2 dξ = 0, (4.9)
By the above discussion, if g is non-trivial then ´R g = 0. Hence θ 2 satisfies

J(θ 2 ) ≡ 1 -B ˆ∞ -∞ χ |ξ| < λ -β 4π 2 Aξ 2 + θ 2 dξ = 0. Since J : (0, ∞) → R is smooth, J (X) > 0, lim X→0 J(X) = -∞ and lim X→∞ J(X) = 1, the function J
has a unique positive root, which we denote by θ 2 0 . One can check by direct substitution and the condition J(θ 2 0 ) = 0, that any multiple of

f0(ξ) ≡ f λ (ξ; θ0) = χ(|ξ| < λ -β ) × (4π 2 Aξ 2 + θ 2 0 ) -1 (4.10) satisfies L0[θ0] f0(ξ) = 0 .
The approximation to θ0(λ), (4.7), is obtained as follows. Let θ 2 0 denote the unique solution of J(θ 2 0 ) = 0 and θ0 its positive square root. Then,

1 B = ˆR χ |ξ| < λ -β 4π 2 A ξ 2 + θ 2 0 dξ = ˆR 1 + (χ |ξ| < λ -β -1) 4π 2 A ξ 2 + θ 2 0 dξ = 1 2 √ A θ0 + ˆR χ |ξ| < λ -β -1 4π 2 Aξ 2 + θ 2 0 dξ . (4.11)
The last term can be bounded as follows:

ˆR 1 -χ |ξ| < λ -β 4π 2 Aξ 2 + θ 2 0 dξ = ˆ|ξ|≥λ -β dξ 4π 2 A ξ 2 + θ 2 0 ≤ ˆ|ξ|≥λ -β dξ 4π 2 A ξ 2 ≤ λ β 2π 2 A . (4.12)
Relations (4.11), (4.12), after rearrangement of terms, yield (4.7).

Finally, let us turn to the asymptotic expression for F -1 f0 (x) given in (4.8). By residue computation, one has

g(ξ) = -2α0 4π 2 |ξ| 2 + α 2 0 = B 4π 2 A|ξ| 2 + B 2

4A

. It follows that

sup x∈R F -1 f0 (x) - 1 B g(x) ≤ f0 - 1 B g L 1 ≤ ˆR χ |ξ| < λ -β 4π 2 Aξ 2 + θ 2 0 - 1 4π 2 A|ξ| 2 + B 2 4A dξ ≤ ˆR χ |ξ| < λ -β 1 4π 2 Aξ 2 + θ 2 0 - 1 4π 2 A|ξ| 2 + B 2 4A dξ + ˆR 1 -χ |ξ| < λ -β 4π 2 Aξ 2 + B 2 4A dξ.
A bound on the second term follows from (4.12). The first term is easily bounded, using (4.7), by C(A, B)λ β , with some constant C(A, B) > 0. Estimate (4.8) follows, and the proof of Lemma 4.1 is now complete.

We shall also require a result on the solvability of the inhomogeneous equation

L0[θ0] ϕ (ξ) = h(ξ), (4.13)
where L0[θ0] is defined in (4.4).

Lemma 4.2. The equation (4.13) is solvable if and only if h is such that χ |ξ| < λ -β h(ξ) = h(ξ) and satisfies the orthogonality condition f0, h

L 2 (R) = 0, (4.14)
where f0, displayed in (4.6), spans the kernel of L0[θ0]. In that case, 1. any solution of the inhomogeneous equation (4.13) is of the form

ϕ(ξ) ≡ (C + h(ξ)) f0(ξ) ≡ (C + h(ξ)) χ |ξ| < λ -β 4π 2 Aξ 2 + θ 2 0 , (4.15)
for some constant C.

2. The unique solution of (4.13) such that ´R ϕ = 0 is obtained by choosing C = 0:

ϕ(ξ) ≡ h(ξ) f0(ξ). (4.16)
Proof. The solvability condition χ |ξ| < λ -β h(ξ) = h(ξ) is straightforward, and (4.14) is obtained by taking the inner product of (4.13) with f0, and using that L0[θ0] is symmetric, and

L0[θ0] f0 = 0.
To show that (4.15) solves the inhomogeneous equation (4.13) we simply insert the function (4.15) into (4.13), and use the properties: L0(θ0) f0 = 0 and f0, h

L 2
= 0. This gives

L0[θ0] ϕ (ξ) = (4π 2 Aξ 2 + θ 2 0 ) h(ξ) f0(ξ) -B χ |ξ| < λ -β ˆ∞ -∞ h(η) f0(η)dη = (4π 2 Aξ 2 + θ 2 0 ) χ |ξ| < λ -β h(ξ) 4π 2 Aξ 2 + θ 2 0 -B χ |ξ| < λ -β f0, h L 2 (R) = h(ξ).
The converse clearly holds by Lemma 4.1, since the difference of solutions of the inhomogeneous equation solves the homogeneous equation (4.4). Finally, using the orthogonality condition f0, h

L 2 = 0, one has that ´R ϕ = C ´R f0 = 0 if and only if C = 0.

A perturbation result for L 0

As discussed in the introduction, our strategy is to obtain a reduction of the eigenvalue problem for H Q+λV to an eigenvalue problem (the bifurcation equation) for functions supported at energies near the band-edge. These reduced equations have a general form which we study in this section. Let Z1 and Z2 denote Banach spaces with Z1, Z2 ⊂ L 1 loc . Assume that for any (f, g) ∈ Z1 ×Z2,

f, g L 2 f Z 2 g Z 1 , f g Z 2 f Z 2 g L ∞ ,
and

(1 + ξ 2 ) -1 f Z 2 f Z 1 . (4.17)
Furthermore, we also assume that f0 ∈ Z1 ∩ Z2, where θ 2 0 , f0 is the unique normalized solution of the homogeneous equation L0[θ] f = 0; see Lemma 4.1.

Remark 4.3. In order to prove Theorems 3.1 and 3.4, we shall apply Lemma 4.4, below, with

• Case Q ≡ 0: (Z1, Z2) = L ∞ , L 1 in the case Q = 0; and • Q non-trivial, 1-periodic: (Z1, Z2) = L 2,-1 , L 2,1
, where L 2,s is the space of locally integrable functions such that

F L 2,s ≡ (1 + |ξ| 2 ) s/2 F L 2 (R ξ ) < ∞.
It is straightforward to check that such spaces satisfy (4.17), and f0 ∈ Z1 ∩ Z2.

We seek a solution of the equation:

L0[θ] f = R f , (4.18)
where L0(θ) is the operator defined in (4.4) and the mapping f → R f is linear and satisfies the following properties: Assumptions R α,β : There exist constants α > 0, β > 0 and CR > 0 such that for any f ∈ Z2

χ |ξ| < λ -β R f (ξ) = R f (ξ) , and R f Z 1 ≤ CRλ α f Z 2 . (4.19)
In the above setting we have the following Lemma 4.4. Let (θ 2 0 , f0(ξ)) be the solution of L0(θ0) f0 = 0, as defined in Lemma 4.1, where A, B and β > 0 are fixed. Let R : f ∈ Z2 → Z1 be a linear mapping satisfying assumptions R α,β displayed in (4. [START_REF] Duchêne | Scattering, homogenization and interface effects for oscillatory potentials with strong singularities[END_REF], where Z1, Z2 satisfy (4.17). Then there exists λ0 > 0 such that for any 0 < λ < λ0, the following holds:

1. There exists a unique solution θ, f (ξ) ∈ R + × Z2 of the equation (4.18), such that

f -f0 Z 2 ≤ Cλ α , ˆ∞ -∞ f (ξ) -f0(ξ) dξ = 0,
with C = C(A, B, CR, β), independent of λ.

Moreover, one has

f (ξ) = χ |ξ| < λ -β f (ξ) and θ 2 -θ 2 0 ≤ Cλ α .
Proof. Our strategy is to use a fixed point argument. We seek a solution (θ 2 , f ) to (4.18) of the form

θ 2 ≡ θ 2 0 + θ 2 1 and f ≡ f0 + f1.
Clearly, any solution f of (4.18) satisfies f (ξ) = χ |ξ| < λ -β f (ξ). Therefore, since one has, by definition, f0(ξ) = χ |ξ| < λ -β f0(ξ), it follows that f1(ξ) = χ |ξ| < λ -β f1(ξ). Substitution of these expressions into (4.18) yields

(4π 2 Aξ 2 + θ 2 )χ |ξ| < λ -β f0 + f1 (ξ) -χ |ξ| < λ -β B ˆ∞ -∞ χ |η| < λ -β f0 + f1 (η)dη = R f0 + f1 (ξ).
Rearranging terms yields the following equation for f1, in which θ 2 1 is a parameter to be determined:

L0[θ0] f1 (ξ) = -θ 2 1 f0 + f1 (ξ) + R f0 + f1 (ξ). (4.20) By Lemma 4.2, (4.20) is solvable in L 2 only if the right hand side is L 2 -orthogonal to f0: f0, -θ 2 1 f0 + f1 + R f0 + f1 L 2 = 0.
Solving for θ 2 1 , we obtain

θ 2 1 = f0, R f0 + f1 L 2 f0, f0 L 2 + f0, f1 L 2 . ( 4
L0[θ0] f1 (ξ) = - f0, R f0 + f1 L 2 f0, f0 L 2 + f0, f1 L 2 f0 + f1 (ξ) + R f0 + f1 (ξ). ( 4
f1(ξ) = G( f1)(ξ), (4.23)
where

G( f1)(ξ) ≡ χ |ξ| < λ -β 4π 2 Aξ 2 + θ 2 0   - f0, R f0 + f1 L 2 f0, f0 L 2 + f0, f1 L 2 f0 + f1 (ξ) + R f0 + f1 (ξ)    . (4.24)
We solve the fixed point equation (4.23) by the contraction mapping principle. Once f1 has been obtained, θ 2 1 is determined using (4.21).

Introduce S = f ∈ Z2 : f Z 2 ≤ CH λ α , for some fixed CH > 0 . (4.25)
Note that S is a closed subset of the Banach space Z2. We next show that there exists λ0 > 0 such that for all 0 < λ < λ0: G : S → S and G is a contraction mapping. As a consequence, it will follow that for 0 < λ < λ0, there is a unique solution f1 ∈ S of the equation f1 = G( f1) and therefore of (4.22). Moreover, f1 λ α by definition of S, and one can check:

ˆR f1 = ˆR G( f1) = ˆR f0(ξ)   - f0, R f0 + f1 L 2 f0, f0 L 2 + f0, f1 L 2 f0 + f1 (ξ) + R f0 + f1 (ξ)    dξ = 0.
It then remains to obtain an estimate of θ1 2 = θ0 2 -θ 2 . From (4.21), one has

θ 2 1 ≤ f0, R f0 + f1 L 2 1 f0, f0 L 2 + f0, f1 L 2 λ α ,
where we used (4.17) and (4. [START_REF] Duchêne | Scattering, homogenization and interface effects for oscillatory potentials with strong singularities[END_REF], and the fact that for λ sufficiently small, f0, f0

L 2 ≥ c > 0,
where c is independent of λ. Lemma 4.4 is proved.

Proof that G : S → S is a contraction mapping: The result will follow from the two following claims, proved below:

Claim 4.5. There exists CH = C θ0, A, CR, f0 Z 2 > 0 such that G(0) Z 2 ≤ 1 2 CH λ α .
Claim 4.6. There exists λ0 > 0 such that if 0 ≤ λ < λ0, then G( f1) -G( f2)

Z 2 ≤ 1 2 f1 -f2 Z 2 . It follows that G maps S ≡ f ∈ Z2 : f Z 2 ≤ CH λ α into S since G(f ) Z 2 ≤ G(f ) -G(0) Z 2 + G(0) Z 2 ≤ 1 2 f -0 Z 2 + 1 2 CH λ α ≤ CH λ α .
Therefore, by Claim 4.6, G : S → S is a contraction mapping.

Proof of Claim 4.5: By definition, one has

G(0)(ξ) ≡ χ |ξ| < λ -β 4π 2 Aξ 2 + θ 2 0 -f0, R f0 L 2 f0(ξ) + R f0 (ξ) .
It follows, from our assumptions (4.17) on functional spaces (Z1, Z2):

G(0) Z 2 χ |ξ| < λ -β 4π 2 Aξ 2 + θ 2 0 L ∞ f0, R f0 L 2 f0 Z 2 + χ |ξ| < λ -β (1 + ξ 2 ) 4π 2 Aξ 2 + θ 2 0 L ∞ R f0 1 + | • | 2 Z 2 R f0 Z 1 f0 2 Z 2 + R f0 Z 1 . (4.26)
Claim 4.5 is now obvious, using the smallness hypothesis on the operator R, (

Proof of Claim 4.6: Let us decompose the mapping G as follows:

G( f1 -f2) = χ |ξ| < λ -β 4π 2 Aξ 2 + θ 2 0 - f0, R f0 + f1 L 2
f0, f0

L 2 + f0, f1 L 2 f0 + f1 (ξ) + f0, R f0 + f2 L 2 f0, f0 L 2 + f0, f2 L 2 f0 + f2 (ξ) + R f1 -f2 (ξ) ≡ S1[ f1](ξ) -S1[ f2](ξ) 4π 2 Aξ 2 + θ 2 0 + χ |ξ| < λ -β R f1 -f2 (ξ) 4π 2 Aξ 2 + θ 2 0 .
The following estimate follows from our assumptions (4.17) on the spaces (Z1, Z2):

G( f1 -f2) Z 2 ≤ S1[ f1](ξ) -S1[ f2](ξ) 4π 2 Aξ 2 + θ 2 0 Z 2 + χ |ξ| < λ -β R f1 -f2 (ξ) 4π 2 Aξ 2 + θ 2 0 Z 2 χ |ξ| < λ -β 4π 2 Aξ 2 + θ 2 0 L ∞ S1[ f1] -S1[ f2] Z 2 + χ |ξ| < λ -β (1 + ξ 2 ) 4π 2 Aξ 2 + θ 2 0 L ∞ R f1 -f2 1 + | • | 2 Z 2 S1[ f1] -S1[ f2] Z 2 + R f1 -f2 Z 1 . (4.27)
The second term in (4.27) is estimated using assumptions R α,β , (4.19):

R f1 -f2

Z 1 ≤ CRλ α f1 -f2 Z 2 . (4.28)
Let us now turn to the first term in (4.27).

S1[ f1] -S1[ f2] = - f0, R f0 + f1 L 2
f0, f0

L 2 + f0, f1 L 2 f0 + f1 + f0, R f0 + f2 L 2
f0, f0

L 2 + f0, f2 L 2 f0 + f2 = - f0, R f1 -f2 L 2 f0 + f1 f0, f0 L 2 + f0, f1 L 2 - f0, R f0 + f2 L 2 f1 -f2 f0, f0 L 2 + f0, f1 L 2 -f0, R f0 + f2 L 2 f0 + f2    1 f0, f0 L 2 + f0, f1 L 2 - 1 f0, f0 L 2 + f0, f2 L 2    = I + II + III. (4.29)
The result is a consequence of the following estimates:

f0, g L 2 ≤ C f0 Z 1 g Z 2 ≤ C1 g Z 2 , f0, R (g) L 2 ≤ C f0 Z 2 R (g) Z 1 ≤ C2λ α g Z 2 .
with C1 = C f0 Z 1 and C2 = C2 f0 Z 2 , CR . Using the above, one checks that for sufficiently small λ,

I Z 2 C2λ α f1 -f2 Z 2 ( f0 Z 2 + CH λ α ), II Z 2 C2λ α ( f0 Z 2 + CH λ α ) f1 -f2 Z 2 , III Z 2 C1 f1 -f2 Z 2 C2λ α ( f0 Z 2 + CH λ α ) 2 .
Thus if C1λ α < 1/2, one has

S1[ f1] -S1[ f2] Z 2 ≤ I Z 2 + II Z 2 + III Z 2 λ α f1 -f2 Z 2 . ( 4.30)
Plugging (4.28) and (4.30) into (4.27), it follows the existence of a constant, C0 > 0, such that

G( f1) -G( f2) Z 2 ≤ C0λ α f1 -f2 Z 2 . Thus for 0 < λ < λ0 ≤ C -1 α 0
, we obtain a contraction and Claim 4.6 is proved.

5 Proof of Th'm 3.1; Edge bifurcations for -∂ 2

x + λV (x)

In this section we prove Theorem 3.1, the special case: Q ≡ 0 of our main result, Theorem 3.4. In this case we study the bifurcation of solutions to the eigenvalue problem

-∂ 2 x + λV (x) ψ λ (x) = E λ ψ λ (x), ψ λ ∈ L 2 (R) (5.1)
into the interval (-∞, 0), the semi-infinite spectral gap of H0 ≡ -∂ 2

x , for V a spatially localized potential, and λ > 0 sufficiently small. Here, the Floquet-Bloch eigenfunctions are exponentials. Hence, calculations are more straightforward and error bounds on the approximations are sharper.

Near and far frequency decomposition

Taking the Fourier transform of (5.1) yields

4π 2 ξ 2 -E λ ψ λ (ξ) + λ ˆR V (ξ -ζ) ψ λ (ζ) dζ = 0. (5.2)
We shall study (5.2) via the equivalent system for the near frequency components: { ψ λ (ξ) : |ξ| < λ r } and far frequency components:

{ ψ λ (ξ) : |ξ| ≥ λ r } of ψ λ .
Let r be a parameter, chosen to satisfy: 0 < r < 1. Recall the cutoff functions, χ and χ, introduced in (1.17) and 1 = χ λ r (ξ) + χ λ r (ξ). Multiplying (5.2) by this identity we get

0 = 4π 2 |ξ| 2 -E λ χ λ r + χ λ r (ξ) ψ λ (ξ) + λ ˆ∞ -∞ χ λ r + χ λ r (ξ) V (ξ -ζ) χ λ r + χ λ r (ζ) ψ λ (ζ)dζ.
Introduce notation for the near-and far-frequency components of ψ λ :

ψnear(ξ) ≡ χ λ r (ξ) ψ λ (ξ) and ψ far (ξ) ≡ χ λ r (ξ) ψ λ (ξ). (5.3)
Then, the eigenvalue equation is equivalent to the following coupled system:

4π 2 |ξ| 2 -E λ ψnear(ξ) + λχ λ r (ξ) ˆ∞ -∞ V (ξ -ζ) ψnear(ζ) + ψ far (ζ) dζ = 0, (5.4)
4π 2 |ξ| 2 -E λ ψ far (ξ) + λχ λ r (ξ) ˆ∞ -∞ V (ξ -ζ) ψnear(ζ) + ψ far (ζ) dζ = 0. (5.5)
In what follows we shall set E λ = -λ 2 θ 2 , where θ = θ(λ) is expected to be O(1) as λ ↓ 0. This anticipates that the bifurcating eigenvalue, E λ , will be real, negative and O(λ 2 ).

Analysis of the far frequency components

We view (5.5) as an equation for ψ far , depending on "parameters" ( ψnear; λ). The following proposition studies the mapping ( ψnear; λ) → ψ far . Proposition 5.1. Let ψnear ∈ L 1 . There exists λ0 > 0, such that for 0 < λ < λ0, the following holds. Set E λ ≡ -λ 2 θ 2 , with |θ| ≤ πλ r-1 , r ∈ (0, 1) . There is a unique solution ψ far = ψ far ψnear; λ of the far frequency equation (5.5). The mapping ( ψnear; λ) → ψ far ψnear; λ maps

L 1 (R) × R to L 1 (R)
and satisfies the bound:

ψ far L 1 ≤ C( V L ∞ ) λ 1-r ψnear L 1 .
(5.6)

Proof. We seek to solve (5.5) for ψ far as a functional of ψnear. Since |ξ| ≥ λ r , with 0 < r < 1, and |θ| ≤ πλ r-1 , we have 4π 2 ξ 2 -E λ = 4π 2 ξ 2 + λ 2 θ 2 ≥ 3π 2 λ 2r , which is bounded away from zero for any fixed λ > 0. Dividing (5.5) by 4π 2 ξ 2 -E λ = 4π 2 ξ 2 + λ 2 θ 2 , we obtain that (5.5) is equivalent to the equation:

I + T λ ψ far (ξ) = -T λ ψnear (ξ) , (5.7)
where

T λ g (ξ) ≡ ˆζ K λ (ξ, ζ) g(ζ) dζ and K λ (ξ, ζ) ≡ λ χ λ r (ξ) 4π 2 ξ 2 + λ 2 θ 2 V (ξ -ζ) .
We next show that the integral operator T λ , viewed as an operator from L 1 to L 1 has small norm, for λ small. This implies the invertibility of I + T λ and the assertions of Proposition 5.1.

Let g ∈ L 1 . One has

T λ g L 1 ≤ C λ ˆ|ξ|≥λ r 1 4π 2 ξ 2 + λ 2 θ 2 dξ V L ∞ g L 1 λ 1-r V L ∞ g L 1 .
Thus T λ is bounded from L 1 to L 1 with norm bound:

T λ L 1 →L 1 λ 1-r V L ∞ . For r ∈ (0, 1), T λ L 1 →L 1 → 0 as λ → 0. Therefore I + T λ is invertible, for λ sufficiently small. Moreover, ψ far L 1 = I + T λ -1 T λ ψnear L 1 ≤ I + T λ -1 L 1 →L 1 T λ L 1 →L 1 ψnear L 1 ,
which implies the bound (5.6). Proposition 5.1 is proved.

Analysis of the near frequency components

Now that we have constructed ψ far as a functional of ψnear and λ (Proposition 5.1), it is possible to treat (5.4), for λ small, as a closed equation for a low frequency projected eigenstate, ψnear(ξ; λ), and corresponding eigenvalue E λ . Substitution of ψ far = ψ far [ ψnear, λ] into (5.4) yields:

4π 2 |ξ| 2 -E λ ψnear(ξ) + λχ λ r (ξ) ˆζ V (ξ -ζ) ψnear(ζ)dζ + λχ λ r (ξ) R(ξ) = 0, (5.8)
where R is defined by

R(ξ) ≡ ˆζ V (ξ -ζ) ψ far [ ψnear, λ](ζ) dζ .
(5.9)

Recall that ψ far [ ψnear, λ] is in L 1 , and of size O λ 1-r ψnear L 1 by Proposition 5.1.

Our next goal is, via appropriate expansion, reorganization and scaling, to re-express (5.8) as a simple leading order asymptotic equation plus controllable corrections. The terms in (5.8) are supported in the near (low) frequency regime. Note that for |ξ| < λ r and |ζ| < λ r we have

|ξ -ζ| ≤ |ξ| + |ζ| < 2λ r . Taylor expansion of V (ξ -ζ) gives V (ξ -ζ) = V (0) + (ξ -ζ) V (η)
, for some η = η(ζ, ξ) such that |η| < 2λ r . Using this expansion in the second term of (5.8) yields

4π 2 |ξ| 2 -E λ ψnear(ξ) + λχ λ r (ξ) V (0) ˆζ ψnear(ζ)dζ = λχ λ r (ξ)R ψnear; λ (ξ), (5.10)
where R ψnear; λ ≡ R1 + R2, with

R1(ξ) ≡ -R(ξ) = - ˆζ V (ξ -ζ) ψ far [ ψnear, λ](ζ) dζ , R2(ξ) ≡ -ˆζ(ξ -ζ) V (η) ψnear(ζ) dζ .
We now introduce the scaled near-frequency Fourier component, Φ λ , by

ψnear(ξ; λ) = 1 λ Φ λ ξ λ , (5.11)
Note that ψnear(•; λ)

L 1 = 1 λ Φ λ • λ L 1 = Φ λ L 1
.

(5.12)

We also denote E λ = -λ 2 θ 2 , and restrict to θ = θ(λ) satisfying the constraint in the hypotheses of Proposition 5.1. Substitution of (5.11) into (5.10), defining ξ = λξ and dividing by λ yields the following rescaled near-frequency equation:

4π 2 |ξ | 2 + θ 2 Φ λ (ξ ) + χ λ r-1 (ξ ) V (0) ˆζ Φ λ (ζ )dζ = χ λ r-1 (ξ )R Φ λ (ξ ) (5.13)
where

R Φ λ (ξ ) ≡ R ψnear; λ (λξ ) ≡ R 1 (ξ ) + R 2 (ξ ), with R 1 (ξ ) ≡ - ˆζ V (λξ -ζ) ψ far [ ψnear, λ](ζ) dζ , (5.14)
R 2 (ξ ) ≡ -ˆζ(λξ -ζ) V (η) ψnear(ζ) dζ = -λ ˆζ(ξ -ζ ) V (η) Φ λ (ζ )dζ .
(5.15) Equation (5.13) is in the form of the class of equations to which Lemma 4.4 applies. We shall use Lemma 4.4 to obtain a non-trivial eigenpair solution ( Φ λ , θ(λ) ) of (5.13). Toward verification of the hypotheses of Lemma 4.4, we next bound the right hand side of (5.13).

Proposition 5.2. Let V be such that V W 1,∞ ≡ V L ∞ + V L ∞ < ∞ .
Then, the right hand side of the rescaled near-frequency equation (5.13) satisfies the bound

χ λ r-1 (ξ)R Φ λ L ∞ ≤ C V W 1,∞ λ 1-r + λ r Φ λ L 1 . (5.16)
Proof. We proceed by estimating each term individually. Estimation of R 1 (ξ ), given by (5.14): By Proposition 5.1, one has

ψ far [ ψnear, λ] L 1 (R) ≤ C V L ∞ λ 1-r ψnear L 1 (R) .
(5.17)

Plugging (5.17) into (5.14), and making use of (5.12), we have

R 1 L ∞ = ˆR V (λξ -ζ) ψ far [ ψnear, λ](ζ) dζ L ∞ ξ ≤ V L ∞ ψ far [ ψnear, λ] L 1 ≤ C( V L ∞ ) λ 1-r Φ λ L 1 .
Estimation of R 2 (ξ ), given by (5.15): We have the bound

χ λ r-1 (ξ )R 2 L ∞ = χ λ r-1 (ξ ) ˆζ λ(ξ -ζ ) V (η) Φ λ (ζ )dζ L ∞ ξ ≤ 2λ r V L ∞ Φ λ L 1 , using that Φ λ (ζ ) = χ λ r-1 (ζ ) Φ λ (ζ ), so that |ξ -ζ | ≤ 2λ r-1 . Proposition 5.2 is proved.
Remark 5.3. We expect that by using a higher order Taylor approximation of V (ξ -ζ) in the second term of equation (5.8), it should be possible to obtain a variant of Proposition 5.2 with a bound which is higher order in λ. This would require a higher order variant of Lemma 4.4.

Completion of the proof

We now prove Theorem 3.1 by an application of Lemma 4.4 to equation (5.13), using the remainder estimate of Proposition 5.2.

Proof of Theorem 3.1. We construct ψ λ , solution to (5.2) as ψ λ = ψ far + ψnear, where ψ far , ψnear satisfy (5.4)-(5.5). The far-frequency component, ψ far , is uniquely determined by ψnear and λ sufficiently small; see Proposition 5.1. Now set ψnear(ξ) ≡ 1 λ Φ λ ξ λ . Since V ∈ W 1,∞ , Proposition 5.2 implies that the rescaled near-frequency equation (5.13) can be written as

4π 2 |ξ | 2 + θ 2 Φ λ (ξ ) + χ λ r-1 (ξ ) V (0) ˆζ Φ λ (ζ )dζ = χ λ r-1 (ξ )R( Φ λ )(ξ ), (5.18)
with

R(u) L ∞ ≤ C λ α u L 1 , where α = min(1 -r, r) and C = C( V W 1,∞ ).
From now on, we set r = 1/2 = α as this yields optimal estimates. Applying Lemma 4.4 to (5.18) with A = 1, -B = V (0) = ´R V (assumed to be negative), we deduce that there exists a solution θ 2 , Φ λ of the rescaled nearfrequency equation (5.18), satisfying

Φ λ -f0 L 1 ≤ C λ 1 2 and |θ 2 -θ 2 0 | ≤ C λ 1 2 . (5.19)
Here θ 2 0 (λ), f0 is the unique (normalized) solution of the homogeneous equation

L 0,λ θ0, f0 = (4π 2 ξ 2 + θ 2 ) f0 + χ |ξ| < λ -1 2 V (0) ˆR χ |η| < λ -1 2 f0(η)dη = 0,
as described in Lemma 4.1. Thus ψnear(ξ) = 1 λ Φ λ ξ λ and E λ = -λ 2 θ 2 (λ) are well-defined. In conclusion, the eigenpair solution to (5.2) (i.e. (3.1)) , (E λ , ψ λ ), is uniquely determined by

E λ ≡ -λ 2 θ 2 (λ),
and ψ λ ≡ F -1 ( ψnear + ψ far ).

Estimate (3.2), the small λ expansion of the eigenvalue E λ , follows from (5.19). The approximation, (3.3), of the corresponding eigenstate, ψ λ = ψnear + ψ far , is obtained as follows. First, by (5.19) we have

ψnear(η) -λ χ λ 1/2 (η) 4π 2 |η| 2 + λ 2 θ 2 0 L 1 = Φ λ -f0 L 1 λ 1/2 .
(5.20)

The high frequency components are small, as is seen from the following calculation:

λ χ λ 1/2 (η) 4π 2 |η| 2 + λ 2 V (0) 2 L 1 ≤ λ ˆ|η|≥λ 1/2 dη 4π 2 |η| 2 λ 1/2 . (5.21)
Finally, from Proposition 5.1, one has (with r = 1/2)

ψ far L 1 ≤ C ( V L ∞ ) λ 1/2 ψnear L 1 , (5.22)
and ψnear L 1 = Φ λ L 1 → f0 L 1 (as λ → 0). Altogether, (5.20), (5.21) and (5.22) yield

ψ λ -F -1 λ 1 4π 2 | • | 2 + λ 2 θ 2 0 L ∞ ≤ ψ λ -λ 1 4π 2 | • | 2 + λ 2 θ 2 0 L 1 λ 1/2 .

Note, by residue computation, that

F -1 (4π 2 | • | 2 + λ 2 θ 2 0 ) -1 = 1 2 (λθ0) -1 exp(-λθ0|x|), with θ0 = -1 2 ´R V > 0 .
Thus estimate (3.3) holds. This completes the proof of Theorem 3.1.

6 Proof of Th'm 3.4; Edge bifurcations of -∂

2 x + Q + λV Let Q(x) denote a non-trivial, continuous, 1-periodic function, Q(x + 1) = Q(x).
In this section we study the bifurcation of solutions to the eigenvalue problem

-∂ 2 x + Q(x) + λV (x) ψ λ (x) = E λ ψ λ (x), ψ ∈ L 2 (R) (6.1)
into the spectral gaps of -∂ 2 x + Q(x). We proceed by the same general approach of Section 5. That is, by appropriate spectral localization, in this case by applying the Gelfand-Bloch transform, we reduce (6.1) to an equivalent near-frequency eigenvalue problem supported on frequencies lying near a spectral band edge of -∂ 2

x + Q(x).

Near and far frequency components

We take the Gelfand-Bloch transform of (6.1) and get

-(∂x + 2πik) 2 ψ λ (x; k) + Q(x) ψ λ (x; k) + λ V ψ λ ∼ (x; k) = E λ ψ λ (x; k), (6.2)
where

V ψ λ ∼ (x; k) = n∈Z e 2πinx V ψ λ ∧ (k + n) = n∈Z e 2πinx V ψ λ (k + n).
Here, the quasi-momentum, k, varies over the interval (-1/2, 1/2].

As in Section 5, we express ψ in terms of its near-and far-frequency components around a band edge E b * (k * ), for fixed b * and k * :

ψ λ = ψnear + ψ far = T -1 ψnear(k)p b * (x; k) + T -1 ∞ b=0 ψ far,b (k)p b (x; k) , (6.3)
where we define, for b = 0, 1, . . . :

ψnear(k) ≡ χ (|k -k * | < λ r ) T b * {ψ λ }(k) = χ (|k -k * | < λ r ) p b * (•, k), ψ λ (•, k) L 2 ([0,1]) , ψ far,b (k) ≡ χ (|k -k * | ≥ λ r δ b * ,b ) T b {ψ λ }(k) = χ (|k -k * | ≥ λ r δ b * ,b ) p b (•, k), ψ λ (•, k) L 2 ([0,1])
, where δi,j denotes Kronecker's delta function and r a parameter chosen to satisfy r > 0. Equivalently, one has

ψ λ (x) = ˆ1/2 -1/2 ψnear(k)u b * (x; k) + ∞ b=0 ψ far,b (k)u b (x; k) dk.
Recall that {p b (x; k)} b≥0 form a complete orthonormal set in L 2 per ([0, 1]), and satisfy

-(∂x + 2πik) 2 + Q(x) p b (x; k) = E b (k)p b (x; k), x ∈ [0, 1], p b (x + 1; k) = p b (x; k) . (6.4)
Therefore, taking the inner product of (6.2) with p b (x; k), and using self-adjointness of the operator -(∂x + 2πik) 2 + Q as well as the identity (6.4), yields

E b (k) -E λ p b (•, k), ψ λ (•, k) L 2 ([0,1]) + λ p b (•, k), V ψ λ ∼ (•, k) L 2 ([0,1]) = 0. (6.5)
or equivalently, using notation (2.5),

E b (k) -E λ T b ψ λ (k) + λT b V ψ λ (k) = 0. (6.6)
We can now decompose equation (6.5) into near-and far-frequency equations, around E b * (k * ), the edge of the b * -th band of the continuous spectrum. The coupled equations for ψnear and ψ far read:

E b * (k) -E λ χ (|k| < λ r ) p b * (•, k), ψ λ (•, k) L 2 ([0,1]) (6.7) + λχ (|k| < λ r ) p b * (•, k), [V (ψnear + ψ far )] ∼ (•, k) L 2 ([0,1]) = 0,
and for b ∈ N:

E b (k) -E λ χ (|k| ≥ λ r δ b * ,b ) p b (•, k), ψ λ (•, k) L 2 ([0,1]) (6.8) + λχ (|k| ≥ λ r δ b * ,b ) p b (•, k), [V (ψnear + ψ far )] ∼ (•, k) L 2 ([0,1]) = 0.
Equivalently, we write the near and far frequency equations in the form

E b * (k) -E λ ψnear(k) + λχ (|k| < λ r ) (T b * {V ψnear} (k) + T b * {V ψ far } (k)) = 0, (6.9)
E b (k) -E λ ψ far,b (k) + λχ (|k| ≥ λ r δ b * ,b ) (T b {V ψnear} (k) + T b {V ψ far } (k)) = 0. (6.10)
Equations (6.9) and (6.10) are, for the case of non-trivial periodic potentials, Q(x), the analogues of (5.4)-(5.5).

Analysis of the far frequency Floquet-Bloch components

In this section we study the far frequency equation (6.10). We will show that we can write it in terms of the near frequency solution and will determine a bound on the far solution in terms of the near solution. The next result is therefore the analogue of Proposition 5. There exists λ0 > 0, such that for 0 < λ < λ0, the following holds. Set

E λ = E * -λ 2 θ 2 , θ ≤ λ r-1 1 2 |∂ 2 k E b * (0)| 1/2 , 0 < r < 1 2 . ( 6

.11)

Then for any ψnear ∈ L 2 (R), there is a unique solution ψ far [ψnear, λ] ∈ L 2 (R) of the far-frequency system (6.10). The mapping (ψnear; λ) → ψ far maps L 2 (R) × (0, λ0) to H 2 (R) and ψ far satisfies the bound

ψ far [ψnear; λ] H 2 (R) ≤ C ( V L ∞ ) λ 1-2r ψnear L 2 (R) .
(6.12) Remark 6.2. Recall that we have assumed

(1 + |x|)V (x) ∈ L 1 (R) and V ∈ L ∞ .
It is in the proof of the bound (6.12) that we have used V ∈ L ∞ . We believe it possible to work under the milder assumption

(1 + |x|)V (x) ∈ L 1 (R).
In this case, we would bound ψ far in H 1 (R) and the analysis that would follow would be a bit more technical. We leave this an exercise.

Proof. We begin by showing that there exists λ0 > 0 such that for all 0 < λ < λ0, there is a constant C1 > 0 such that For |k| ≤ λ0, we approximate E b * (k) by a Taylor expansion. In particular, since

|E b * (k) -E * | ≥ C1λ 2r , λ r ≤ |k| ≤ 1/2 , (6.13) |E b (k) -E * | ≥ C1, b = b * , |k| ≤ 1/2 . ( 6
E b * (k) is smooth for k near k * = 0, ∂ k E b * (0) = 0 and ∂ 2 k E b * (0) = 0, we have E b * (k) -E b * (0) -1 2 ∂ 2 k E b * (0) k 2 = O(|k| 3
). Therefore, we can choose λ0 > 0 sufficiently small so that for all λ ≤ λ0 we have

|E b * (k) -E b * (0)| ≥ 1 3 ∂ 2 k E b * (0) λ 2r , for all λ ≤ |k| ≤ λ0 . (6.16)
It follows from (6.15) and (6.16) that for sufficiently small λ0 > 0,

1 2 ≥ |k| ≥ λ > 0 =⇒ |E b * (k) -E * | ≥ min 1 3 ∂ 2 k E b * (0) λ 2r , C(λ0) .
Thus if

E λ = E * -λ 2 θ 2 , θ ≤ λ r-1 1 2 |∂ 2 k E b * (0)| 1/2
, then for 0 < λ < λ0 sufficiently small, there is a positive constant C1, such that

|E b * (k) -E λ | ≥ C1λ 2r .
(6.17) By (6.13) and (6.14), the far-frequency system, (6.10), may be re-written as

ψ far,b (k) + λ χ (|k| ≥ λ r δ b * ,b ) E b (k) -E λ T b {V ψ far } (k) = -λ χ (|k| ≥ λ r δ b * ,b ) E b (k) -E λ T b {V ψnear} (k), b ≥ 0. ( 6
.18) We wish to rewrite this equation in terms of ψ far (x). In order to do so, we multiply (6.18) by u b (x; k) = p b (x; k)e 2πikx , sum over b ≥ 0 and integrate with respect to k ∈ (-1/2, 1/2]. This yields

(I + K λ ) ψ far (x) = -(K λ ψnear) (x), (6.19)
where we define

(K λ g) (x) ≡ ˆ1/2 -1/2 b≥0 λ χ (|k| ≥ λ r δ b * ,b ) E b (k) -E λ T b {V g} (k)p b (x; k)e 2πikx dk.
We next show that the operator K λ , viewed as an operator from L 2 to H 2 has small norm, for λ small. Let g ∈ L 2 . Using Proposition 2.4, one has

K λ g 2 H 2 K λ g 2 X 2 = ˆ1/2 -1/2 b≥0 (1 + b 2 ) 2 |T b {K λ g} (k)| 2 dk = λ 2 ˆ1/2 -1/2 b≥0 (1 + b 2 ) 2 χ (|k| ≥ λ r δ b * ,b ) |E b (k) -E λ | 2 |T b {V g} (k)| 2 dk.
Now, by (6.17), for |k| ≥ λ r one has E b * (k) -E λ -1 ≤ C1λ -2r , and recall 0 < r < 1/2. For b = b * , we use Weyl asymptotics to write

(E b (k) -E λ ) -1 ∼ (b 2 -E * ) -1 ∼ (b 2 + 1) -1 . We therefore have K λ g 2 H 2 λ 2 ˆ1/2 -1/2 b≥0 |T b {V g} (k)| 2 dk + λ 2-4r ˆ1/2 -1/2 (1 + b * 2 ) 2 χ (|k| ≥ λ r ) |T b * {V g} (k)| 2 dk λ 2-4r (V g) ∼ 2 X 0 λ 2-4r V 2 L ∞ g 2 L 2
. Thus, since r ∈ (0, 1/2), one can choose λ0 > 0 such that if 0 < λ < λ0, then K λ L 2 →H 2 < 1. In particular, K λ is a contraction from L 2 to L 2 , and therefore I + K λ is invertible. The existence and uniqueness of ψ far ∈ L 2 (R) solution to (6.10) is now given through (6.19). Moreover, one has

ψ far H 2 = (I + K λ ) -1 (K λ ψnear) H 2 ≤ (I + K λ ) -1 H 2 →H 2 K λ L 2 →H 2 ψnear L 2 λ 1-2r V L ∞ ψnear L 2 ,
which implies the bound (6.12). The proof of Proposition 6.1 is complete.

Analysis of the near frequency Floquet-Bloch component

With the properties of the map ψnear → ψ far [ψnear, λ] now understood via Proposition 6.1, we now view and study (6.9) as a closed eigenvalue problem for (E λ , ψnear):

E b * (k) -E λ ψnear(k) + λχ λ r (k)T b * {V ψnear} (k) + λ χ λ r (k)T b * {V ψ far [ψnear; λ] } (k) = 0.
(6.20) Equation (6.20) is localized in the region |k| < λ r , 0 < r < 1/2. By careful expansion and rescaling of (6.20) we shall obtain an equation, which at leading order in λ, is a perturbation of the general class of equations to which Lemma 4.1 applies. The size of the perturbation is estimated in Proposition 6.6 and the perturbed equation is then solved by applying Lemma 4.4.

In Lemmata 6.3, 6.4 and 6.5 we expand the first two terms in (6.20) about k * = 0 using Taylor's Theorem, making explicit the leading and higher order contributions. Lemma 6.3. Denote E λ = E * -λ 2 θ 2 = E b * (0) -λ 2 θ 2 , as in Proposition 6.1. There exists k such that |k | < λ r , and

E b * (k) -E λ ψnear(k) = 1 2 ∂ 2 k E b * (0) k 2 + λ 2 θ 2 ψnear(k) + λR0 ψnear; λ (k, k ), where R0 ψnear; λ (k, k ) = 1 λ 1 4! k 4 ∂ 4 k E b * (k ) ψnear(k). (6.21)
Proof. Taylor expanding E b * (k) about k * = 0 to fourth order and making use of

E λ = E b * (0)-λ 2 θ 2 and ∂ j k E b * (0) = 0 for j = 1, 3, one obtains E b * (k) -E λ = 1 2 ∂ 2 k E b * (0)k 2 + λ 2 θ 2 + 1 4! ∂ 4 k E b * (k )k 4
, which is equivalent to (6.21). Lemma 6.4. One can decompose Above, we used that T -1 commutes with multiplication by a 1-periodic function of x, and that when acting on a function which is localized near k = 0, and which does not depend on x, T -1 is equivalent to the standard inverse Fourier transform; see Section 2.

T b * {V ψnear} (k) = p b * (•; 0), p b * (•; 0)T V F -1 ψnear (•; k) L 2 ([0,1]) + R1 ψnear; λ (k), with
R1 ψnear; λ (k) = p b * (•; 0), T {V E1} (•, k) L 2 ([0,1]) + p b * (•; k) -p b * (•; 0), T {V ψnear} (•, k) L 2 ([0,1]) , ( 6
The proof of Lemma 6.4 is now straightforward.

We next give a precise expression of the leading order term in Lemma 6.4.

Lemma 6.5. One can decompose

p b * (•; 0), p b * (•; 0)T V F -1 { ψnear} (•; k) L 2 ([0,1]) (6.26) = ˆ∞ -∞ |p b * (x; 0)| 2 V (x)dx ˆ∞ -∞ ψnear(l)dl + R2 ψnear (k), with R2 ψnear (k) = ˆ∞ -∞ dx |p b * (x; 0)| 2 V (x) ˆ∞ -∞ e 2iπ(l-k)x -1 ψnear(l) dl. (6.27)
Proof. By the definition of T , one has

T V F -1 { ψnear} (x; k) = n∈Z e 2πinx F V F -1 { ψnear} (k + n) = n∈Z e 2πinx ˆ∞ -∞ V (k + n -l) ψnear(l) dl.
Since |k| < λ r and ψnear(l) is localized on |l| < λ r , the leading order term is obtained when replacing V (k + n -l) with V (n). The first term of (6.26) now follows from the identity:

n∈Z p b * (•; 0), p b * (•; 0)e 2πin• L 2 ([0,1]) V (n) = ˆ1 0 |p b * (x; 0)| 2 n∈Z e 2πinx V (n)dx = n∈Z ˆ1 0 |p b * (x; 0)| 2 V (x + n)dx = ˆ∞ -∞ |p b * (x; 0)| 2 V (x)dx.
Here, we used the Poisson summation formula and that x → p b * (x; 0) is 1-periodic.

Similarly, one has

n∈Z p b * (•; 0), p b * (•; 0)e 2πin• L 2 ([0,1]) V (n + k -l) = ˆ∞ -∞ |p b * (x; 0)| 2 e 2iπ(l-k)x V (x)dx.
This completes the proof of Lemma 6.5.

The rescaled closed equation. Using Lemmata 6.3, 6.4 and 6.5, one can express the near frequency equation (6.20) as follows:

( 1 2 ∂ 2 k E b * (0)k 2 + λ 2 θ 2 ) ψnear(k) + λ χ (|k| < λ r ) ˆ∞ -∞ |p b * (x; 0)| 2 V (x)dx ˆ∞ -∞ ψnear(l)dl = -λχ (|k| < λ r ) R [ψnear; λ] (k), (6.28)
where

R [ψnear; λ] (k) ≡ T b * {V ψ far } + R0 + R1 + R2.
Seeking to extract the dominant and higher order terms in λ, we introduce the scaled nearfrequency components:

ψnear(k) = 1 λ Φ λ k λ = 1 λ Φ λ (κ) , where k = λ κ . (6.29)
Expressing (6.28) in terms of Φ λ and κ we obtain, after dividing out by λ,

1 2 ∂ 2 k E b * (0)κ 2 + θ 2 χ λ r-1 (κ) Φ λ (κ) + ˆR |p b * (•; 0)| 2 V χ λ r-1 (κ) ˆR χ λ r-1 (η) Φ λ (η)dη = -χ |κ| < λ r-1 R [ψnear; λ] (λκ) ≡ R( Φ λ ). (6.30) Equation (6.30) is of the form L0[θ] Φ λ (κ) = R( Φ λ )
, where L0[θ] is given by (4.1) with parameters

A = 1 8π 2 ∂ 2 k E b * (0) , B = -ˆR |p b * (x; 0)| 2 V (x)dx , and β = 1 -r .
In order to solve (6.30) via Lemma 4.4 we need a bound on R( Φ λ ) of the form (4.19).

Proposition 6.6. Assume that V is such that

(1 + | • |)V (•) ∈ L 1 and V ∈ L ∞ . Then R( Φ λ ),
defined in (6.30), satisfies the bound

R( Φ λ ) L 2,-1 = χ | • | < λ r-1 R [ψnear; λ] (λ•) L 2,-1 ≤ Cλ α(r) Φ λ L 2,1 . (6.31)
where α(r) = max 1 2 -2r, 2r, r+1

2

. The constant C depends on

(1+|•|)V L 1 , V L ∞ as well as sup |k|<λ r p b * (•; k) L ∞ , sup |k|<λ r n∈Z p b * (•; k), e 2πin• L 2 ([0,1]) , sup |k|<λ r ∂ 4 k E b * (k) , sup |k|<λ r ∂ k p b * (•; k) L ∞ ,
and is finite by Lemmata 2.3 and 2.5.

Proof of Proposition 6.6. Recall that R(λκ), the right hand side of (6.30) has the form

R [ψ far [ψnear; λ] , ψnear; λ] (λκ) = χ |κ| < λ r-1 T b * {V ψ far } (λκ) + R0 ψnear; λ (λκ, k ) + R1 ψnear; λ (λκ) + R2 ψnear (λκ) ≡ (I) + (II) + (III) + (IV ) . (6.32)
We proceed by estimating each of the terms: (I), (II), (III) and (IV ).

(I) Estimation of χ |κ| < λ r-1 T b * {V ψ far } (λκ): We have χ | • | < λ r-1 T b * {V ψ far } (λ•) 2 L 2,-1 = ˆ∞ -∞ χ |κ| < λ r-1 1 + κ 2 |T b * {V ψ far } (λκ)| 2 dκ ≤ T b * {V ψ far } 2 L ∞ .
We now consider T b * {V ψ far } (•) in detail. By definition, one has

T b * {V ψ far } (k) = p b * (•; k), T {V ψ far } (•, k) L 2 ([0,1]) = p b * (•; k), n∈Z e 2πin• ˆ∞ -∞ V (k + n -l) ψ far (l)dl L 2 ([0,1]) = n∈Z p b * (•; k), e 2πin• L 2 ([0,1]) ˆ∞ -∞ V (k + n -l) (1 + |l| 2 ) 1/2 (1 + |l| 2 ) 1/2 ψ far (l)dl.
Moreover,

ˆ∞ -∞ V (k + n -l) (1 + |l| 2 ) 1/2 (1 + |l| 2 ) 1/2 ψ far (l)dl ≤ V L ∞ ψ far H 2 λ 1-2r ψnear L 2 λ 1-2r ψnear L 2 = λ 1-2r λ -1 2 Φ λ L 2
, where we used Proposition 6.1, definition (6.29) and, by Proposition 2.4,

ψnear 2 L 2 = T -1 ψnear(k)p b * (x; k) 2 L 2 ψnear(k)p b * (x; k) 2 X 0 = ˆ1/2 -1/2 | ψnear(k)| 2 dk = ψnear 2 L 2 . (6.33)
Finally, it follows

T b * {V ψ far } L ∞ ≤ λ 1 2 -2r C Φ λ L 2,1 . (6.34)
with

C = C V L ∞ , V L ∞ , sup |k|<λ r n∈Z p b * (•; k), e 2πin• L 2 ([0, 1])
.

(II) Estimation of χ |κ| < λ r-1 R0 ψnear; λ (λκ, k ), given in (6.21): We have (constants implicit)

χ | • | < λ r-1 λ 2 (•) 4 Φ λ (•) 2 L 2,-1 (R) = λ 4 ˆ∞ -∞ κ 8 1 + κ 2 χ |κ| < λ r-1 Φ λ (κ) 2 dκ = λ 4 ˆ∞ -∞ κ 8 (1 + κ 2 ) 2 χ |κ| < λ r-1 (1 + κ 2 ) Φ λ (κ) 2 dκ λ 4 sup |κ|<λ r-1 κ 8 (1 + κ 2 ) 2 Φ λ 2 L 2,1 λ 4r Φ λ 2 L 2,1 . Therefore, χ |κ| < λ r-1 R0 ψnear; λ (λκ, k ) L 2,-1 ≡ χ |κ| < λ r-1 1 4! ∂ 4 k E b * (k )λ 2 κ 4 Φ λ (κ) L 2,-1 λ 2r sup |k |<λ r |∂ 4 k E b * (k )| Φ λ L 2,1 . (6.35)
(III) Estimation of χ |κ| < λ r-1 R1 ψnear; λ (λκ), given in (6.22): Recall

R1 ψnear; λ (k) = p b * (•; 0), T {V E1} (•, k) L 2 ([0,1]) + p b * (•; k) -p b * (•; 0), T {V ψnear} (•, k) L 2 ([0,1]) . (6.36)
where

E1 ≡ T -1 ψnear(k) p b * (x; k) -p b * (x; 0) .
Let us first obtain an estimate on E1. Using Taylor expansion of p b * (x; •) around 0, one has

|E1(x)| = ˆ1/2 -1/2 e 2πikx ψnear(k) p b * (x; k) -p b * (x; 0) dk ≤ sup x∈R,|k |<λ r |∂ k p b * (x; k )| ˆ∞ -∞ |kχ (|k| < λ r ) ψnear(k)| dk ≤ λ sup |k |<λ r ∂ k p b * (•; k ) L ∞ ˆ∞ -∞ |κχ |κ| < λ r-1 Φ λ (κ)| dκ ≤ λ sup |k |<λ r ∂ k p b * (•; k ) L ∞ ˆ|κ|<λ r-1 κ 2 1 + κ 2 dκ 1/2 Φ λ L 2,1 ≤ 2λ 1+r 2 sup |k |<λ r ∂ k p b * (•; k ) L ∞ Φ λ L 2,1
, so that we deduce

E1 L ∞ ≤ 2λ 1+r 2 sup |k |<λ r ∂ k p b * (x; k ) L ∞ Φ λ L 2,1 . (6.37)
Estimation of the first term of (6.36) is as follows. One has

χ |κ| < λ r-1 p b * (•; 0), T {V E1} (•, λκ) L 2 ([0,1]) 2 L 2,-1 κ = ˆ∞ -∞ χ |κ| < λ r-1 1 + κ 2 p b * (•; 0), T {V E1} (•, λκ) L 2 ([0, 1]) 2 dκ.
Turning to the integrand of the above expression, we rewrite the inner product

p b * (•; 0), T {V E1} (•, λκ) L 2 ([0,1]) = ˆ1 0 T {p b * (•; 0)E1(•)V (•)} (x; λκ) = ˆ1 0 n∈Z e 2πinx F {p b * (•; 0)E1(•)V (•)} (λκ + n) dx = F {p b * (•; 0)E1(•)V (•)} (λκ),
where we used that p b * (x; 0) is 1-periodic, so that it commutes with T , and the Poisson summation formula. It follows that

p b * (•; 0), T {V E1} (•, λκ) L 2 ([0,1]) ≤ p b * (•; 0)E1(•)V (•) L 1 ≤ E1 L ∞ ˆ|p b * (x; 0)||V (x)| dx.
Using (6.37), one deduces

χ |κ| < λ r-1 p b * (•; 0), T {V E1} (•, λκ) L 2 ([0,1]) L 2,-1 κ ≤ Cλ 1+r 2 Φ λ L 2,1 , (6.38)
with

C = C(sup |k|<λ r ∂ k p b * (•; k) L ∞ , ´|p b * (x; 0)||V (x)| dx).
The last term in (6.36) is estimated as follows. Note that

p b * (•; λκ) -p b * (•; 0), T {V ψnear} (•, λκ) L 2 ([0,1]) = ˆ1 0 p b * (x; λκ) -p b * (x; 0) n∈Z e 2πinx F{V ψnear}(λκ + n)dx = ˆ1 0 p b * (x; λκ) -p b * (x; 0) n∈Z (V ψnear)(x + n)e -2πi(λκ+n)x dx ≤ ˆ∞ -∞ p b * (x; λκ) -p b * (x; 0) V (x)ψnear(x) dx ≤ λκ sup |k |<λ r ∂ k p b * (•; k ) L ∞ ψnear L ∞ V L 1 ,
where we used the Poisson summation formula along with the periodicity of p b * (x; λκ) -p b * (x; 0) and its Taylor expansion as |λκ| < λ r . Now, note that

ψnear L ∞ = T -1 ψnear(k)p b * (x; k) L ∞ ≤ sup |k|<λ r p b * (•; k) L ∞ ˆλr -λ r | ψnear(l)| dl and ˆ∞ -∞ | ψnear(l)| dl = ˆ∞ -∞ | Φ λ (η)| dη = ˆ∞ -∞ 1 (1 + η 2 ) 1/2 (1 + η 2 ) 1/2 | Φ λ (η)| dη ≤ C Φ λ L 2,1 . It follows χ |κ| < λ r-1 p b * (•; λκ) -p b * (•; 0), T {V ψnear} (•, λκ) L 2 ([0,1]) L 2,-1 κ ≤ Cλ Φ λ L 2,1 ˆκ2 χ |κ| < λ r-1 1 + κ 2 1/2 λ 1+r 2 Φ λ L 2,1 , (6.39)
with

C = C sup |k|<λ r p b * (•; k) L ∞ , sup |k |<λ r ∂ k p b * (•; k ) L ∞ , V L 1 .
Estimates (6.38) and (6.39) yield

χ |κ| < λ r-1 R1[ ψnear](λκ) L 2,-1 ≤ C Φ λ L 2,1 λ 1+r 2 . (6.40) with C = C sup |k|<λ r p b * (•; k) L ∞ , sup |k|<λ r ∂ k p b * (•; k) L ∞ , V L 1 . (IV ) Estimation of χ |κ| < λ r-1 R2[ ψnear](λκ), given in (6.27): Recall R2 ψnear (k) = ˆ∞ -∞ dx |p b * (x; 0)| 2 V (x) ˆ∞ -∞ e 2iπ(l-k)x -1 ψnear(l)dl.
We now use that e 2iπ(l-k)x -1 ≤ 2π|l -k||x|. It follows

R2 ψnear (λκ) ≤ 2πλ ˆ∞ -∞ dx |p b * (x; 0)| 2 |x V (x)| ˆ∞ -∞ |κ -η|| Φ λ (η)|dη.
We therefore define

I(κ) = -χ |κ| < λ r-1 ˆ∞ -∞ |κ -η|χ |η| < λ r-1 | Φ λ (η)|dη. (6.41)
The integral, I(κ), is bounded in L 2,-1 (R) as follows:

I 2 L 2,-1 ≤ ˆ∞ -∞ χ |κ| < λ r-1 1 + κ 2 ˆ|η|<λ r-1 |κ -η| 2 1 + η 2 dηdκ Φ λ 2 L 2,1 = Φ λ 2 L 2,1 ˆκ ˆη |κ -η| 2 (1 + κ 2 )(1 + η 2 ) χ |κ| < λ r-1 χ |η| < λ r-1 dκdη.
One easily checks that

ˆκ ˆη |κ -η| 2 (1 + κ 2 )(1 + η 2 ) χ |κ| < λ r-1 χ |η| < λ r-1 dκdη λ r-1 ,
so that one obtains eventually

χ |κ| < λ r-1 R2[ ψnear](λκ) L 2,-1 ≤ C Φ λ L 2,1 λ r+1 2 , (6.42) with C = C sup |k|<λ r p b * (•; k) L ∞ , xV (x) L 1 x .
Altogether, (6.34), (6.35), (6.40), and (6.42) yield the estimate of Proposition 6.6.

Completion of the proof of Theorem 3.4

We now prove Theorem 3.4 by an application of Lemma 4.4 to equation (6.30), where the remainder is estimated in Proposition 6.6.

Proof of Theorem 3.4. We seek E λ ≡ E b * (0) -λ 2 θ 2 and ψ λ of the form

ψ λ = ψnear + ψ far = T -1 ψnear(k)p b * (x;k) + T -1 ∞ b=0 ψ far,b (k)p b (x; k) = ˆ1/2 -1/2 ψnear(k)u b * (x; k) + ∞ b=0 ψ far,b (k)u b (x; k) dk.
where ψnear, ψ far satisfy equations (6.9)-(6.10); see Section 6.1. By application of Proposition 6.1, one has that ψ far is uniquely defined as a function of ψnear and λ, and that ψ far [ψnear; λ] H 2 ≤ λ 1-2r ψnear L 2 . Then, defining Φ λ as in (6.29), one has

ψnear(k) = 1 λ Φ λ k λ = 1 λ Φ λ (κ) , k = λκ. (6.43)
By Proposition 6.6, the rescaled (from (6.9)) near-frequency equation (6.30) can be written as

1 2 ∂ 2 k E b * (0)κ 2 + θ 2 χ λ r-1 (κ) Φ λ (κ) + χ λ r-1 (κ) ˆR |p b * (•; 0)| 2 V ˆR χ λ r-1 (η) Φ λ (η)dη = -χ |κ| < λ r-1 R Φ λ (κ), (6.44) with R Φ λ L 2,-1 ≤ Cλ α(r) Φ λ L 2,1
, and α(r) = max( 1 2 -2r, 2r, r+1 2 ). From now on, we set r = 1/8, α = 1/4, which yield optimal estimates. Applying Lemma 4.4 with β = 1 -r = 7/8,

A = 1 8π 2 ∂ 2 k E b * (0) and B = - ˆ∞ -∞ |u b * (x; 0)| 2 V (x)dx
assumed to be positive , (6.45)

we deduce that there exists a solution θ 2 , Φ λ of the rescaled near-frequency equation (6.44), satisfying Φ λ -f0 L 2,1 ≤ C λ , so that one can set Let us now estimate I2(x) and I3(x). One has

E2 ≡ - B 2 4A = - ´∞ -∞ |u b * (x; k * )| 2 V (x)dx
|I2(x)| ≡ u b * (x; 0) ˆR χ |ξ| < λ -7 8 Φ λ -f0 (ξ)e 2πiλξx dξ ≤ |p b * (x; 0)| ˆR χ |ξ| < λ -7 8 (1 + |ξ| 2 ) 1/2 (1 + |ξ| 2 ) 1/2 Φ λ (ξ) -f0(ξ) dξ ≤ C p b * (•; 0) L ∞ Φ λ -f0 L 2,1 ≤ C(A, B) p b * (•; 0) L ∞ λ 1/4 , (6.49)
where the last inequality comes from (6.46). Similarly,

|I3(x)| ≡ ˆR χ |ξ| < λ -7 8 Φ λ (ξ) e 2πiλξx (λξ)∂ k p b * (x; k ) dξ ≤ λ sup |k |<λ 1-7/8 ∂ k p b * (•; k ) L ∞ ˆR χ |ξ| < λ -7 8 |ξ| Φ λ (ξ) dξ ≤ C sup |k |<λ 1/8 ∂ k p b * (•; k ) L ∞ Φ λ L 2,1
λ. (6.50) By (6.48), (6.49) and ( 6.50), one has

ψnear = I1(x) + I2(x) + I3(x) = 2 B u b * (x; 0) exp -λB 2A |x| + ψrem(x),
with ψrem L ∞ λ 1/4 . Finally, let us note that by Sobolev embeddings, one has

ψ far L ∞ ≤ ψ far H 2 ≤ Cλ 1-1/4 ψnear L 2 = Cλ 1/2-1/4 Φ λ L 2 ≤ Cλ 1/4 ,
where we use Proposition 6.1 with r = 1/8, and (6.33). It follows that ψ λ = ψnear + ψ far satisfies

sup x∈R ψ λ (x) - 1 B u b * (x; 0) exp(λα0|x|) ≤ Cλ 1/4 , with α0 = - B 2A .
Since ψ λ is defined up to a multiplicative constant, (3.11) holds. This completes the proof of Theorem 3.4.

A General properties of E b (k) and derivatives

∂ j k E b (k * ), where E b (k * ) is

the endpoint of a spectral band

To make our discussion more self-contained, we prove Lemma 2.2, which concerns the spectrum of the eigenvalue problem, for E fixed,

-∂ 2 x + Q(x) ψ(x; E) = Eψ(x; E), Q(x + 1) = Q(x), (A.1) Lemma A.1 (Lemma 2.2). Assume E b (k * ) is an endpoint of a spectral band of -∂ 2 x + Q(x)
, which borders on a spectral gap; see (2.3). Then k * ∈ {0, 1/2} and the following results hold:

1. E b (k * ) is a simple eigenvalue of the eigenvalue problem (2.1). 3.

∂ k E b (k * ) = 0, ∂ 3 k E b (k * ) = 0; 4. b even: ∂ 2 k E b (0) > 0, ∂ 2 k E b (1/2) < 0; b odd: ∂ 2 k E b (0) < 0, ∂ 2 k E b (1/2) > 0;
The proof of Lemma A.1 is a consequence of the following result, concerning the problem (A.1), and which is proved in the first two chapters of [START_REF] Eastham | The spectral theory of periodic differential equations[END_REF] and part I of [START_REF] Magnus | Hill's Equation[END_REF]. Similarly, E2m+1(k) decreases continuously from G2m+1 to F2m+1 as k increases continuously from 0 to 1/2, and as k decreases continuously from 0 to -1/2. This proves claim 2 . We now turn to part 1 . Let E b (k * ) correspond to a band edge, that is if there exists a gap between the b th band and the closest consecutive one. Without loss of generality, we assume E b (k * ) to be the lowermost edge of an even band, for example G2m in

D(E) = D(E b (k * )) + D (E b (k * ))(E -E b (k * )) + 1 2 D (E b (k * ))(E -E b (k * )) 2 + O (E -E b (k * )) 3 = -2 + 1 2 D (E b (k * ))(E -E b (k * )) 2 + O (E -E b (k * )) 3 . Since D (E b (k * )) < 0, we have D(E b (k * ) -δ) ≈ 2 + (1/2)D (E b (k * ))δ 2 < 2,
which is a contradiction of (A.9). Therefore part 1 is proven and we have that at the band edges, E b (k * ), the derivative of the discriminant is nonzero,

dD dE (E b (k * )) = 0. (A.10)
To see the first identity in part 3 , note that differentiating

D(E b (k)) = 2 cos(2πk) with respect to k yields -4π sin(2πk) = dD dE (E b (k)) × dE b dk (k). Using (A.10), we conclude that dE b dk (k) = 0 if and only if k = 0 or k = 1/2.
To prove part 4 , we differentiate D(E b (k)) twice with respect to k and evaluate at k * :

-8π 2 cos(2πk * ) = d 2 dk 2 (D • E b )(k * ) = D (E b (k * )) dE b dk 2 (k * ) + D (E b (k * )) d 2 E b dk 2 (k * ).
Therefore, by I , II , and (A.10) we conclude 4 . Similarly, to show the second identity of part 3 , we differentiate once more D(E b (k)) with respect to k:

16π 3 sin(2πk) = D (E b (k)) d 3 E b dk 3 (k) + 3D (E b (k)) d 2 E b dk 2 d E b dk (k) + D (E b (k)) dE b dk 3 (k). Evaluated at k * , we have 0 = D (E b (k)) d 3 E b dk 3 (k)
, which concludes the proof of Lemma 2.2 once we again note (A.10).

B Regularity of

k → E b (k) and k → u b (x; k)
In this section we give a self-contained discussion of the regularity with respect to k of the Floquet-Bloch eigenvalues and eigenstates.

Consider the k-pseudo-periodic eigenvalue problem for each k ∈ (-1/2, 1/2]:

-∂ 2 x + Q(x) u(x; k) = Eu(x; k), u(x + 1; k) = e 2πik u(x; k) (B.1)
Introducing the Floquet-Bloch phase explicitly via u(x; k) = e 2πikx p(x; k), we obtain the equivalent formulation

HQ(k)p(x; k) = -(∂x + 2πik) 2 + Q(x) p(x; k) = Ep(x; k), p(x + 1; k) = p(x; k) . (B.2)
For each k ∈ (-1/2, 1/2], the eigenvalue problem (B.2) (equivalently (B.1)) has a discrete sequence of eigenvalues E0

(k) ≤ E1(k) ≤ E2(k) ≤ • • • ≤ En(k) ≤ • • • .
It can be proved, using the min-max characterization of eigenvalues of a self-adjoint operator that the maps k → E b (k), b = 0, 1, . . . , are locally Lipschitz continuous. A proof based on standard perturbation follows from results in [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]. An elementary proof is given in Appendix A of [START_REF] Fefferman | Wavepackets in honeycomb structures and two-dimensional Dirac equations[END_REF].

In Then, there exists ρ > 0 such that for all complex k in a complex disc centered at k * , Bρ(k * ) = {k ∈ C : |k -k * | < ρ}, the following holds:

1. k → E b (k) is analytic on Bρ.
µ1 -8π 2 k * -4π 2 κ + 4πiκ p(•; k * ), ∂xη1(•) = 0. (B.12) Let R(E * )Π ⊥ = (HQ(k * ) -E * ) -1 Π ⊥ . Then, η1(x; µ1, κ) = 4πi I -κR(E * )Π ⊥ 4πi(∂x + 2πik ) -4π 2 κ + µ1 -1 R(E * ) Π ⊥ ∂xp(•; k * ) , (B.13)
where we take |κ| < ρ, with ρ chosen so that the Neumann series for the operator on the right hand side of (B.13) converges. Note that the mapping

(µ1, κ) → η1(x; µ1, κ)
is an analytic map from {(µ1, κ) : |µ1| < 1, |κ| < ρ } to H 2 per (R). Subsitution of (B.13) into (B.12) gives the scalar equation

G(µ1, κ) = 0 , (B .14)
where

G(µ1, κ) = µ1 -8π 2 k * -4π 2 κ + 4πiκ p(•; k * ), ∂xη1 (

C The bootstrap: proof of Corollary 3.6

We give the proof of Corollary 3.6, on the refined expansion of the bifurcation of eigenvalues of

HQ + λV = -∂ 2 x + Q(x) + λV (x), for Q(x) periodic. Corollary 3.3, in the case of Q(x) ≡ 0, is obtained along the same lines, using p(x; k) = 1, E(k) ≡ 4π 2 k 2 for k ∈ R, etc.
Proof of Corollary 3.6. We know, by Theorem 3.4, that there exists (ψ λ , E λ ) solution of the eigenvalue problem (HQ + λV )ψ λ = E λ ψ λ . Moreover, E λ is in the gap of the continuous spectrum of spec(HQ) = spec(H Q+λV ), near an edge E * = E b * (k * ). In the following, we assume that k * = 0 (the case where k * = 1/2 can be treated using the same method).

We next seek an integral equation for ψ λ by applying the resolvent RQ(E λ ) to the differential equation for ψ λ . A construction of the resolvent kernel, RQ(x, y; E λ ), proceeds as follows. Recall the discriminant, D(E), introduced in Appendix A as the trace of the monodromy matrix defined by the linearly independent solutions φj(x; E), j = 1, 2:

D(E) = φ1(1; E) + φ 2 (1; E).
Since E * is a band edge and E λ is in a gap, we have D(E * ) = 2 and D(E λ ) > 2. Therefore, there exists κ = κ(λ) > 0 with

E λ = E(iλκ) = E(-iλκ), D(E λ ) = e 2πλκ + e -2πλκ > 2,
Additionally, we define ψ± ≡ ψ±(x; E λ ), the solutions of

-∂ 2 x + Q(x) ψ± = E λ ψ±, ψ±(x + 1; E λ ) = e ±2πλκ ψ±(x; E λ ).
More precisely, ψ± are defined as

Q , R (1)
Q are bounded:

|R (0) Q (x, y)| + |R (1) Q (x, y)| ≤ Ce -2πλκ|x-y| ≤ C ,
where C is a constant, uniform with respect to λκ.

In order to ease the reading, we postpone the proof of this result to the end of this section, and carry on with the proof of Corollary 3.

. 11 )

 11 which bifurcates from the edge, E * = E b * (k * ), of B b * into a spectral gap. 1. If ∂ 2 k E b * (k * ) > 0 and ´R |u b * (x; k * )| 2 V (x) dx < 0, then µ < 0 and E(λ) lies near the lowermost edge of B b * ; see the center panel of Figure 1. 2. If ∂ 2 k E b * (k * ) < 0 and ´R |u b * (x; k * )| 2 V (x) dx > 0, then µ > 0 and E(λ) lies near the uppermost edge of B b * ; see the right panel of Figure 1.

H 2 b

 2 b * ,eff = -d dy A b * ,eff d dy + B b * ,eff δ(y) , (1.12) with constant effective parameters A b * ,eff and B b * ,eff . Here,A b * ,eff = 1 8π 2 ∂ 2 k E b * (k * ) (1.13)is the inverse effective mass associated to the spectral edge,E * = E b * (k * ), B b * ,eff = ˆR |u b * (x; k * )| 2 V (x) dx ,(1.14)and δ(y) denotes the Dirac delta mass at y = 0. The unique discrete eigenvalue, µ , of the eigenvalue problem: H b * ,eff ψ = µψ is easily seen to be µ = -B

Figure 1 :

 1 Figure 1: Sketch of spectra. Eigenvalues, E b (k), k ∈ (-1/2, 1/2], b = 0, 1, 2, . . . , are displayed in green. The continuous spectrum, is in blue, and discrete eigenvalues are indicated through cross markers. Left panel corresponds spec(H Q ), Q

Proposition 6 . 1 .

 61 1 and facilitates the reduction of the eigenvalue problem to a closed equation for the near-frequency components of the eigenstate. For clarity of presentation and without any loss of generality, we assume henceforth that we are localizing near the lowermost end point of the b * -th band and that k * = 0. Thus, by Lemma 2.2, b * is even, k * = 0, with E b * (0) = E * . N.B. For k * = 0, note that p b (x; k * ) = u b (x; k * ) and we use these expressions interchangeably. For k * = 1/2 one has to distinguish between p b (x; k * ) and u b (x; k * ). Assume b * is even and consider E * = E b * (0) the lowermost edge of the b * -th band.

. 14 ) 2 min λ 0 ≤|k|≤1/ 2 |E

 1422 Note first that(6.14) is an immediate consequence of E * being the endpoint of the (b * ) th spectral gap. To prove (6.13) recall, by Lemma 2.2 that E * = E b * (0), an eigenvalue at the edge of a spectral gap, is simple, and k → E b * (k) -E * is continuous. Therefore, for any λ0, such that 0 < λ0 < 1/b * (k) -E * | ≥ C(λ0) > 0.(6.15)

. 22 )

 22 where E1 ≡ T -1 ψnear(k) p b * (x; k) -p b * (x; 0) . Proof. Let us recall that by definition (6.3), one has ψnear(x) = T -1 ψnear(•)p b * (x; •) . (6.23) Since ψnear(k) = χ (|k| < λ r ) ψnear(k), we decompose: ψnear(x) = T -1 ψnear(•)p b * (x; •) (x) = p b * (x; 0)F -1 { ψnear} + E1(x) (6.24) where E1(x) ≡ T -1 ψnear(•) p b * (x; •) -p b * (x; 0) . (6.25)

8 ˆ∞ -∞ χ |η| < λ - 7 8 2 - 1 / 2

 87212 is a solution of the homogeneous equationL 0,λ θ0, f0 = (4π 2 Aξ 2 + θ 2 ) f0 -B χ |ξ| < λ -7 f0(η)dη = 0,as described in Lemma 4.1. Thus ψnear(ξ) = 1 λ Φ λ ξ λ and E λ = E b * (0)-λ 2 θ 2 (λ) are well-defined (and satisfy the Ansatz of Lemma 6.3), and ψ far is uniquely determined as the solution of (6.10); see Lemma 6.1. It follows that ψ λ (x) ≡ ψ far + ψnear ≡ ψ far + ˆ1/ψnear(k)u b * (x; k) dk (6.47) is well-defined. There remains to prove estimates (3.10) and (3.11). Recalling that E λ = E b * (0) -λ 2 θ 2 , (6.46) implies E λ -(E b * (0) -λ 2 θ 2 0 ) ≤ Cλ 2+1/4 . By Lemma 4.1, one has θ0(λ) -B 2 √ A ≤ C(A, B)λ 7 8

2 12π 2 2 - 1 χ |ξ| < λ -7 8 Φ 8 Φ 8 Φ 8 Φ 8 Φ 1 8 7 8

 22218888817 ∂ 2 k E b * (k * ) ; and estimate (3.10) follows. We now turn to a proof of the eigenfunction approximation (3.11). Recall ψnear(x) ≡ ˆ1/λ (ξ) e 2πiλξx p b * (x; λξ) dξ = ˆR χ |ξ| < λ -7 λ (ξ) e 2πiλξx p b * (x; 0) dξ+ ˆR χ |ξ| < λ -7 λ (ξ) e 2πiλξx (λξ)∂ k p b * (x; k ) dξ = u b * (x; 0) ˆR χ |ξ| < λ -7 8 f0(ξ)e 2πiλξx dξ + u b * (x; 0) ˆR χ |ξ| < λ -7 λ -f0 (ξ)e 2πiλξx dξ + ˆR χ |ξ| < λ -7 λ (ξ) e 2πiλξx (λξ)∂ k p b * (x; k ) dξ = I1(x) + I2(x) + I3(x), with |k | = |k (λξ)| < λ . Now, since χ |ξ| < λ -f0(ξ) = f0(ξ), one has I1(x) ≡ u b * (x; 0)F -1 f0 (λx). (x; 0) F -1 f0 (λx) -1 B exp -λB 2A |x| ≤ C p b * (•; 0) L ∞ λ 7/8 . (6.48) and p b * (•; 0) L ∞ is bounded; see Lemma 2.5.

2 .

 2 b even: E b (0) corresponds to the left (lowermost) end point of the band, E b (1/2) corresponds to the right (uppermost) end point. b odd: E b (0) corresponds to the right (uppermost) end point of the band, E b (1/2) corresponds to the left (lowermost) end point.

Figure 2 : 2 .

 22 Figure 2: Sketch of the discriminant, D(E), and stability bands B b = [G b , F b ].

Figure 2 .

 2 Therefore, for any δ > 0 sufficiently small, D(E b (k * ) -δ) > 2 and D(E b (k * ) + δ) < 2. (A.9) Assume for the sake of contradiction that E b (k * ) is a double eigenvalue, which means, by part V of Theorem A.2, that D (E b (k * )) = 0 and D (E b (k * )) < 0. Now, Taylor expand the discriminant about E b (k * ),

  the present paper, we require a Taylor expansion of the E b (k) near k = k * , for which E b (k * ) is the endpoint of a spectral band, which borders on a spectral gap. By part 5 of Lemma A.1, the eigenvalue E b (k * ) is simple. We prove the following Theorem B.1. Suppose E * is the endpoint of a spectral band of -∂ 2 x + Q(x), which borders on a gap. Thus, E * = E b (k * ) for k * ∈ {0, 1/2} and the corresponding eigenspace of solutions to (B.2) has dimension equal to 1. We denote the normalized eigenfunction by p(x; k * ); ˆ1 0 |p(y; k * )| 2 dy = 1.

2 .

 2 There is a map k → p b (x; k), such that any eigenvector corresponding to E b (k) is a multiple of p b (x; k), where HQ(k)p b (x; k) = E b (k)p b (x; k).3. Moreover, we can choosek → p b (x; k), k ∈ Bρ to be analytic and such that ˆ1 0 |p b (x; k)| 2 dx = 1.Proof of Theorem B.1: Let k = k * +κ, where κ will be chosen to be sufficiently small. The periodic eigenvalue problem (B.2) may be rewritten asHQ(k * )p(x; k * + κ) -4πiκ(∂x + 2πik * ) -4π 2 κ 2 p(x; k * + κ) = E p(x; k * + κ), (B.3) p(x + 1; k * + κ) = p(x; k * + κ) , x ∈ R . (B.4)We seek an eigen-solution of (B.3)-(B.4) in the form p(x; k * + κ) = p(x; k * ) + η(x; κ), (B.5) E(k * + κ) = E * + µ(κ), (B.6) where we assume that η(•; κ) ⊥ p(•; κ). Substitution into (B.3)-(B.4) yields the following equation for η(x; µ, κ): (HQ(k * ) -E * ) η -4πiκ(∂x + 2πik ) -4π 2 κ 2 + µ η = 4πiκ(∂x + 2πik ) -4π 2 κ 2 + µ p(•; k * ) . (B.7) Now, introduce the projection operators Πf = p(•; k * ), f p(x; k * ), and Π ⊥ = I -Π.Applying Π ⊥ to (B.7) yields(HQ(k * ) -E * ) η -Π ⊥ 4πiκ(∂x + 2πik ) -4π 2 κ 2 + µ η = 4πiκ Π ⊥ ∂xp(•; k * ) = 4πiκ∂xp(•; k * ).(B.8) Next, applying Π to (B.7), i.e. taking the inner product of (B.7) with p(•; k * ), yields µ -8π 2 k * κ -4π 2 κ 2 + 4πi κ p(•; k * ), ∂xη(•; µ, κ) = 0. (B.9) We shall now solve (B.7) for η, substitute the result into (B.9) and obtain a closed equation for the eigenvalue correction µ = µ(κ). Let µ = κµ1, η = κη1. (B.10) Equations (B.8) and (B.9) become (HQ(k * ) -E * ) η1 -κΠ ⊥ 4πi(∂x + 2πik ) -4π 2 κ + µ1 η1 = 4πiΠ ⊥ ∂xp(•; k * ), (B.11)

  •; µ1, κ) . (B.15) We now claim that (B.14) can be solved for µ1 = µ1(κ), which is defined and analytic for |κ| < ρ , where 0 < ρ ≤ ρ. If this claim is valid, then η1(x; µ1(κ), κ) is well-defined and analytic in κ for |κ| < ρ and finally p(x; k * + κ) = p(x; k * ) + κη1(x; µ1(κ), κ) (B.16) E(k * + κ) = E * + κµ1(κ) (B.17) are defined and analytic Floquet-Bloch eigensolutions for |κ| < ρ . Now (B.14) is easily solved for µ1 = µ1(κ) via the Implicit Function Theorem. Indeed, we have G(8π 2 k * , 0) = 0 and ∂µ 1 G(µ1, κ)| (8π 2 k * ,0) = 1 = 0. This completes the proof of Theorem B.1.

Q

  ψ±(x) ≡ p b * (x; ∓iκ)e ±2πλκx , with (C.1)-(∂x -2πλκ) 2 + Q(x) p b * (x; iκ) = E λ p b * (x; iκ) , p b * (x + 1; iκ) = p b * (x; iκ). (C.2)which is well-defined for λ small enough, by Theorem B.1. With those definitions, the resolvent operatorRQ(E λ ) = (-∂ 2 x + Q -E λ ) -1 has kernel RQ(x, y; E λ ) =      ψ+(x)ψ-(y) W [ψ±] if y > x, ψ+(y)ψ-(x) W [ψ±] if y < x.whereW [ψ±] ≡ ψ + (x)ψ-(x) -ψ+(x)ψ -(x). Thus, for any bounded function f ,RQ[f ](x; E λ ) = ˆ∞ -∞ RQ(x, y; E λ )f (y) dy, we have (-∂ 2 x + Q -E λ )RQ[f ](x; E λ ) = f .It follows that ψ λ satisfies the integral equation ψ λ (x) + λ ˆR RQ(x, y; E λ )V (y)ψ λ (y) dy = 0. Multiplying by u b * (x; 0)V (x) and integrating along x yields ˆR V (x)u b * (x; 0)ψ λ (x)dx + λ ¨R2 u b * (x; 0)V (x)RQ(x, y; E λ )V (y)ψ λ (y) dx dy = 0. (C.3) We will deduce from (C.3) the precise behavior of κ (and therefore E λ -E b * (0)) as λ tends to zero, using the following Lemma C.1. Let E * = E b * (0) be an edge of the continous spectrum, and let the hypotheses of Theorem 3.4 be satisfied, so that E λ exists. Define RQ(x, y; E λ ) as above. Then for λ > 0 small enough, one has RQ(x, y; E λ ) = u b * (x; 0)u b * (y(x, y) + λκR

6 . 2 ∂ 2 kRemark C. 3 .±

 6223 By Lemma C.1, and since u b * (x; 0) is uniformly bounded (see Lemma 2.5), one has the low-order estimate RQ(x, y; E λ ) -4π ∂ 2 k E(0) u b * (x; 0)u b * (y; 0) 2λκ ≤ C(1 + |x -y| + λκ), (C.5)where we used e -λκ|x-y| -1 ≤ Cλκ|x -y|.Plugging (C.5) into (C.3) and using (1 + |x|)V ∈ L 1 , yields ˆR V (x)u b * (x; 0)ψ λ (x)dx+ 2π κ∂2 k E(0) ¨R2 u b * (x; 0) 2 V (x)u b * (y; 0)V (y)ψ λ (y) dx dy ≤ Cλ(1+λκ).(C.6) Now we use the fact that by Theorem 3.4, one hasψ λ (x) -u b * (x; 0) exp λα0|x| L ∞ λ 1/4 , so that lim λ→0 ´V (x)u b * (x; 0)ψ λ (x)dx = ´V (x)u b * (x; 0) 2 = 0.It follows that for λ sufficiently small, one can divide out ´V (x)u b * (x; 0)ψ λ (x)dx, and deduce from (C.6)κ + 2π ∂ 2 k E(0) ˆR u b * (x; 0) 2 V (x) dx ≤ Cλκ(1 + λκ),from which it follows the low-order estimate of κ:κ + 2π ∂ 2 k E(0) ˆR u b * (x; 0) 2 V (x) dx ≤ Cλ. (C.7)Let us now derive higher order estimates. For anyx, y ∈ R 2 , e -2πλκ|x-y| -1 + 2πλκ|x -y| ≤ 4π 2 λ 2 κ 2 |x -y| 2 , so that one has from Lemma C.1, RQ(x, y; E λ ) -2π ∂ 2 k E(0) u b * (x; 0)u b * (y; 0)(1 -2πλκ|x -y|) λκ -R (0) Q (x, y) ≤ Cλ(1 + |x| 2 + |y| 2 ).(C.8) Plugging (C.8) into (C.3), and using(1 + |x|)V ∈ L 1 , yields ˆub * (x; 0)V (x)ψ λ (x)dx+ 2π ∂ 2 k E(0) 1 κ ¨V (x)u b * (x; 0) 2 (1-2πλκ|x-y|)u b * (y; 0)V (y)ψ λ (y) dx dy + λ 2 ¨V (x)u b * (x; 0)R (0) Q (x, y)V (y)ψ λ (y) dx dy ≤ Cλ 2 . (C.9)Let us now use that by Theorem 3.4,sup x∈R |ψ λ (x) -u b * (x; 0) exp λα0|x| | λ 1/4 , so |ψ λ (x) -u b * (x; 0)| ≤ C(λ 1/4 + λ|x|). Thus (C.9) becomes ˆub * (•; 0)V ψ λ ¨V (x)u b * (x; 0) 2 |x -y|u b * (y; 0) 2 V (y) dx dy + λ 2 ¨V (x)u b * (x; 0)R (0) Q (x, y)V (y)u b * (y; 0) dx dy ≤ Cλ 1+1/4 , (C.10)and one deduces from (C.7) that κ ´ub * (•; 0)V ψ λ -1 + 2π(∂ 2 k E(0)) -1 ≤ Cλ 1/4 . Therefore, multiplying (C.10) by κ ´ub * (•; 0)V ψ λ -1 yields κ + 2π ∂ 2 k E(0) ˆV (x)u b * (x; 0) 2 dx + λ 8π 3 (∂ 2 k E(0)) 2 ¨V (x)u b * (x; 0) 2 |x -y|u b * (y; 0) 2 V (y) dx dy λ 4 ¨V (x)u b * (x; 0)R (0) Q (x, y)V (y)u b * (y; 0) dx dy ≤ Cλ 1+1/4 . (C.11)Finally, we note that since R (0)Q (x, y) = -R(0)Q (y, x) by Lemma C.1, the last term in (C.11) vanishes. Thus the above estimate, together with the following Lemma, completes the proof of Corollary 3.6.Lemma C.2. Let E * = E b * (0) be an edge of the continuous spectrum, and let hypotheses of Theorem 3.4 be satisfied, so that E λ exists. Then for λ small enough, one has E λ = E(iλκ), andE λ -E * = -1 2 λ 2 κ 2 ∂ 2 k E b * (0) + O(λ 4 ) .Proof. We Taylor expand D(E) about E * = E b * (0).D(E) = D(E * ) + D (E * )(E -E * ) + O (E -E * ) 2 = 2 + D (E * )(E -E * ) + O (E -E * ) 2 .(C.12)Let's first apply (C.12) to E = E b * (k) in the spectral band. One hasD(E b * (k)) = e 2πik + e -2πik = 2-4π 2 k 2 +O k 3 . Finally, since ∂ k E b * (0) = ∂ 3 k E b * (0) = 0, one has E b * (k) = E * + 1 2 ∂ 2 k E(0)k 2 + O k 4 . Identifying with (C.12), it follows D (E * ) 1 2 ∂ 2 k E(0) = -4π 2 , thus D (E * ) = -8π E b * (0) . Next let's apply (C.12) to E = E λ , recalling D(E λ ) = e 2πλκ +e -2πλκ = 2+4π 2 λ 2 κ 2 +O λ 4 κ 4 .One has from (3.10) in Theorem 3.4 that E λ -E * = O(λ 2 ), and from (C.7) that κ = O(1). Consequently, (C.12) yields4π 2 λ 2 κ 2 = D (E * )(E λ -E * ) + O(λ 4 ) = -8π 2 ∂ 2 k E b * (0) (E λ -E b * (0)) + O(λ 4 ).Finally, we deduceE λ -E * = -1 2 λ 2 κ 2 ∂ 2 k E b * (0) + O(λ 4) and the lemma is proved.We conclude this section by the proof of Lemma C.1 Proof of Lemma C.1. Let us Taylor-expand ψ±, as defined by (C.1)-(C.2). One has ψ±(x)e ∓2πλκx ≡ p b * (x; ∓iλκ), thusψ+(x)e -2πλκx = p b * (x; 0) -iλκ∂ k p b * (x; 0) -(λκ) 2 2 ∂ 2 κ p(x; 0) + i (λκ) 3 6 ∂ 3 κ p(x; iγ+), (C.13) ψ-(x)e 2πλκx = p b * (x; 0) + iλκ∂ k p b * (x; 0) -(λκ) 2 2 ∂ 2 κ p(x; 0) -i (λκ) 3 6 ∂ 3 κ p(x; iγ-), (C.14) with -λκ ≤ γ+ ≤ 0 ≤ γ-≤ λκ. Note that κ → p b (x; k * +κ) ∈ L 2 (R) is analytic in a complex neighborhood |κ| < κ1. By the equation for p b , κ → p b (x; k * + κ) ∈ H 2 (R)is analytic and thus ∂ 3 k p b (x; k) and ∂x∂ 3 k p b (x; k) are well-defined and uniformly bounded for k near k * and x in any compact set. Since p b * (x; 0) = u b * (x; 0), it follows W [ψ±]RQ(x, y; E λ ) =    u b * (x; 0)u b * (y; 0) + iλκr (0) (x, y; λκ) + (λκ) 2 r (1) + (x, y) e 2πλκ(x-y) if y > x, u b * (y; 0)u b * (x; 0) + iλκr (0) (y, x; λκ) + (λκ) 2 r (1) -(x, y) e 2πλκ(y-x) if y < x. (C.15) with r (0) (x, y; λκ) ≡ p b * (x; 0)∂ k p b * (y; 0) -∂ k p b * (x; 0)p b * (y; 0) = -r (0) (y, x; λκ), (x, y) is bounded, uniformly with respect to λκ. Let us now turn to W [ψ±] ≡ ψ + (x)ψ-(x) -ψ+(x)ψ -(x). From (C.13)-(C.14), one has W [ψ±] = 2λκ 2πp b * (x; 0) 2 -ip b * (x; 0)∂x∂ k p b * (x; 0) + i ∂xp b * (x; 0) ∂ k p b * (x; 0) + (λκ) 3 wr(x; λκ), with wr(x) uniformly bounded, independently of x and λκ. Since W [ψ±] is independent of x, one has W [ψ±] = ´1 0 W [ψ±] dx and thus W [ψ±] = 2λκ ˆ1 0 2πp b * (x; 0) 2 -ip b * (x; 0)∂x∂ k p b * (x; 0) + i ∂xp b * (x; 0) ∂ k p b * (x; 0) dx + (λκ) 3 ˆ1 0 wr(x; λκ) dx. Using that ´1 0 p b * (x; 0) 2 dx = ´1 0 u b * (x; 0) 2 dx = 1, one deduces W [ψ±] = 2λκ 2π + 2i ˆ1 0 p(x; 0)∂x∂ k p(x; 0) dx + O (λκ) 3 . (C.16) Now, let us recall that p b * (x; iκ) satisfies (C.2). Deriving twice with respect to k = iκ, one obtains -(∂x -2πκ) 2 + Q(x) -E(iκ) ∂ 2 k p(x; iκ) = 2∂ k E(iκ)∂ k p(x; iκ) + ∂ 2 k E(iκ)p(x; iκ) -8πi(∂x -2πκ)∂ k p(x; iκ) -8π 2 p(x; iκ) . We now apply this identity at κ = 0, and take the inner product with p b * (x; 0). It follows 0 = ∂ 2 k E(0) -8πi ´1 0 p(x; 0)∂x∂ k p(x; 0) dx -8π 2 . Therefore, (C.16) becomes W [ψ±] = 2λκ ∂ 2 k E(0) 4π + O (λκ) 3 . (C.17) Finally, (C.15) and (C.17) clearly imply (C.4), and Lemma C.1 is proved.

Mathematical preliminariesIn this section we summarize basic results on the spectral theory of Schrödinger operators with periodic potentials defined on R. For a detailed discussion, see for example,[20,

[START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF][START_REF] Magnus | Hill's Equation[END_REF].

Acknowledgements: I.V. and M.I.W. acknowledge the partial support of U.S. National Science Foundation under U.S. NSF Grant DMS-10-08855, the Columbia Optics and Quantum Electronics IGERT NSF Grant DGE-1069420 and NSF EMSW21-RTG: Numerical Mathematics for Scientific Computing. This research was carried out while V.D. was the Chu Assistant Professor of Applied Mathematics at Columbia University.

with solutions which satisfy ψ(x + 1; E) = ρψ(x; E) ρ ∈ C.

Let φ1(x; E) and φ2(x; E) be two linearly independent solutions of (A.1) such that φ1(0; E) = 1, φ2(0; E) = 0,

The functions φ1(x + 1; E) and φ2(x + 1; E) are two other linearly independent solutions to (A.1), so that we can write

Note that the matrix (Aij) is nonsingular. In general, every solution of (A.1) has the form

As we are specifically interested in solutions which satisfy ψ(x + 1; E) = ρψ(x; E), one has the following identity

The solvability condition (A.5) is satisfied for nontrivial c1 and c2 if

Using that the Wronskian, W [φ1, φ2] (x; E) ≡ φ1(x; E)φ 2 (x; E) -φ 1 (x; E)φ2(x; E), is constant with respect to x, one has

Therefore ρ must satisfy ρ 2 -D(E)ρ + 1 = 0, where we define the discriminant

We note that the two solutions of the equation ρ 2 -D(E)ρ + 1 = 0 satisfy |ρ| ≤ 1 if and only if the discriminant |D(E)| ≤ 2. In that case, one can write ρ = e ±2πik , with k ∈ (-1/2, 1/2], and

(A.8)

As |ρ| = 1, ψ(x; E) is a bounded solution to (A.1), and

where {E b (k), p b (x; k)} b≥0 is the eigenpair solution to (2.2), as defined in Section 2.

Let us now rewrite Lemma 2.2 which states some of the properties associated with the stability bands.