Laurent Gosse

Impurity bands and quasi-Bloch waves for a one-dimensional model of modulated crystal

Keywords: Quasi-periodic potential, impurity band, quasi-Bloch wave, quantum chaos, spectral algorithm, deformed crystal, Charge-Density Wave (CDW). 1991 MSC: 81Q05, 81Q20

This paper investigates a simple one-dimensional model of incommensurate "harmonic crystal" in terms of the spectrum of the corresponding Schrödinger equation. Two angles of attack are studied: the first exploits techniques borrowed from the theory of quasi-periodic functions while the second relies on periodicity properties in a higher-dimensional space. It is shown that both approaches lead to essentially the same results; that is, the lower spectrum is splitted between "Cantor-like zones" and "impurity bands" to which correspond critical and extended eigenstates, respectively. These "new bands" seem to emerge inside the band gaps of the unperturbed problem when certain conditions are met and display a parabolic nature. Numerical tests are extensively performed on both steady and time-dependent problems.

Introduction

This article is a step further in the numerical study of electronic motion in a one-dimensional model of "harmonic crystal", that is to say, a modeling in solid-state physics going beyond the rigid periodic lattice picture. Indeed, it consists in assuming that, within the Born-Oppenheimer approximation (the motion of nucleons is much slower than the electron's ones and thus decouples adiabatically), ionic cores are linked together by springs and vibrate like classical coupled oscillators; consult the book [START_REF] Ashcroft | Solid-state physics[END_REF] and the recent paper [START_REF] Gosse | Multiphase semiclassical approximation of the one-dimensional harmonic crystal. I. The periodic case[END_REF] to which we are about to borrow notation.

In [START_REF] Gosse | Multiphase semiclassical approximation of the one-dimensional harmonic crystal. I. The periodic case[END_REF], the following rescaled problem describing the motion of a free electron inside a 1-D lattice of atoms which display single-mode vibrations has been investigated within the * Corresponding Author Email address: l.gosse@ba.iac.cnr.it (Laurent Gosse).

Preprint submitted to Elsevier Science

WKB semi-classical framework [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF][START_REF] Guillot | Semiclassical asymptotics in solid-state physics[END_REF] for the following Schrödinger equation,

iλ∂ t ψ + λ 2 2 ∂ xx ψ = cos x λ -ε sin kx λ -Ω(k)t ψ, ε 1, λ 1 (1)
and under the fundamental assumption that k and Ω(k) = | sin(kπ)| are rational and commensurate. This is perhaps the simplest non-trivial model available which describes the effects of phonons onto electronic conduction and is sometimes referred to as the "modulated crystal" [START_REF] Ashcroft | Solid-state physics[END_REF][START_REF] Emery | One-dimensional fluctuations and chain ordering transformations in Hg 3-δ AsF 6[END_REF][START_REF] De Lange | Incommensurability and recursivity: lattice dynamics of modulated crystals[END_REF][START_REF] De Lange | Electrons in incommensurate crystals: spectrum and localization[END_REF][START_REF] Niu | Quantum adiabatic particle transport[END_REF]. Since the potential in (1) isn't of the form cos(x -1 2 at 2 ), the unitary transform propsed in [START_REF] Peik | Bloch oscillations of atoms, adiabatic rapid passage and monokinetic atomic beams[END_REF] and leading to Bloch oscillations doesn't apply here (see also [START_REF] Niu | Atomic Landau-Zener tunneling and Wannier-Stark ladders in optical potentials[END_REF] for an AC-type modulation). A particular feature pointed out in [START_REF] Sundaram | Wave-packet dynamics in slowly perturbed crystals: gradient corrections and Berry-phase effects[END_REF] and called lattice tracking which refers to the dragging effect of heavy nucleons onto the much lighter electron was easily observed in the numerical results.

The commensurability hypothesis is of critical importance in this global picture in order to keep on applying the classic Bloch decomposition for (1); we assume the reader familiar with this theory and refer only to [START_REF] Ashcroft | Solid-state physics[END_REF][START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF][START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional periodic lattice. I. Homogeneous problems[END_REF][START_REF] Guillot | Semiclassical asymptotics in solid-state physics[END_REF] for details. The first step in this line of thinking is the derivation of "energy bands", which are dispersion relations for electrons around certain levels of excitation. Solids are classified in metals, semi-conductors and insulators depending if the Fermi energy level lies inside a partially filled conduction band or somewhere inside a gap being of moderate or big size, respectively. Generally, semi-classical approximation restricts itself to considering only the propagation of charge carriers in valence and conduction bands: lower ones correspond to inner shells electrons strongly tied to the atomic cores and higher ones are usually empty except in special cases of intense excitation. It has been observed that the use of modern shock-capturing numerical techniques allow for quite satisfying computations of wave packets in increasingly complex environments including possibly self-interactions: see [START_REF] Gosse | Multiphase semiclassical approximation of the one-dimensional harmonic crystal. I. The periodic case[END_REF][START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional periodic lattice. I. Homogeneous problems[END_REF][START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -II. Impurities, confinement and Bloch oscillations[END_REF][START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -III. From ab-initio models to WKB for Schrödinger-Poisson[END_REF][START_REF] Gosse | Finite moment problems and applications to multiphase computations in geometric optics[END_REF]. (left), k = 2 √ 5-1 (right). Abscissas and ordinates correspond to the "quasi-momentum" κ and the energies E in (6) respectively.

However, a simple linearization around the Mathieu (or Hill) potential in [START_REF] Ashcroft | Solid-state physics[END_REF] reveals that in case k and Ω(k) become incommensurate, one has to deal with a quasi-periodic perturbation 1 inside the Schrödinger equation. It is commonly believed that as long as the strength of the perturbation remains small (how small ?), its effect onto the energy bands is negligible and corresponding Bloch states vibrate according to the same wavenumber k, see e.g. [START_REF] Sundaram | Wave-packet dynamics in slowly perturbed crystals: gradient corrections and Berry-phase effects[END_REF] §IV. One could argue that this would be at best a perturbative KAM-type2 result, expressing the fact that the Hamiltonian structure in phase space remains mostly unaltered. We intend to show here that this picture is overmuch simplified. New phenomena can and do occur according to the modulated potential in [START_REF] Ashcroft | Solid-state physics[END_REF]: to convince at once the reader of this fact, we present in Fig. 1 the band structure of (1) for k = 5 3 , k = 2 √ 5-1 (the so-called golden mean) and its inverse inside the Brillouin zone of the classical Mathieu equation, i.e.

[- 1 2 , 1 2 ]. Comparing with Fig. 1 in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional periodic lattice. I. Homogeneous problems[END_REF], hard would it be to see anything they might share!

Looking more carefully at the picture on the right corresponding to the golden mean, one may observe an amazing phenomenon: inside the intervals occupied by the energy bands of the unperturbed Mathieu equation, messy states appear whereas it is exactly inside the "exband gaps" that seemingly new bands show up. How could this be heuristically explained ? First, let us stress the fact that, in case k = 5 3 , the perturbed potential remains periodic in the x variable with t fixed, even if the period has become quite big. Consequently, the real Brillouin zone (like the ones considered previously in [START_REF] Gosse | Multiphase semiclassical approximation of the one-dimensional harmonic crystal. I. The periodic case[END_REF]) is tiny. Hence looking at this scale can give the feeling that these "new bands" look like constant states related to an isolated impurity (this has a lot to do with the "hydrogenic donor" approximation, see [START_REF] Ashcroft | Solid-state physics[END_REF][START_REF] Rehr | Impurity states between two bands[END_REF] and Fig. 9 in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -II. Impurities, confinement and Bloch oscillations[END_REF]). Obviously, when k has become totally irrational, the real Brillouin zone is reduced to a point. Further, one sees on the right picture of Fig. 1 that original (unperturbed) bands have completely collapsed: this is a consequence of a phenomenon called band nesting and beautifully illustrated in [START_REF] Hollingworth | Wave propagation in onedimensional optical quasiperiodic systems[END_REF]. Loosely speaking, it refers to a mechanism which creates smaller and smaller "mini-gaps" inside a given band thus splitting it into a Cantor-like structure, see again [START_REF] De Lange | Electrons in incommensurate crystals: spectrum and localization[END_REF][START_REF] Kohmoto | Critical wave functions and Cantor-set spectrum of a one-dimensional quasicrystal model[END_REF].

In §2.1, we shall review briefly the rigorous mathematical results on eigenfunctions for quasi-periodic 1-D problems which followed the seminal article by Dinaburg and Sinai, [START_REF] Dinaburg | The one-dimensional Schrödinger equation with quasi-periodic potential Funkt[END_REF]. They all rely on difficult KAM arguments and are mostly concerned with bounded solutions of the quasi-periodic Hill's equation y + (E -q(x)) = 0, for small or high E, see [START_REF] Moser | An extension of a result by Dinaburg and Sinai on quasiperiodic potentials Comment[END_REF][START_REF] Johnson | The rotation number for almost periodic potentials[END_REF][START_REF] Fröhlich | Localization for a class of one-dimensional quasi-periodic Schrödinger operators[END_REF][START_REF] Surace | The Schrödinger equation with a quasi-periodic potential[END_REF][START_REF] Eliasson | Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation[END_REF]. However, our situation here is slightly different as we are dealing with a periodic Hill equation with a (possibly small) incommensurate perturbation added: our main theoretical references should be therefore [START_REF] Broer | Resonances tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation[END_REF][START_REF] Gentile | Stability for the quasi-periodically perturbed Hill's equation[END_REF], even if it doesn't exactly fit in [START_REF] Ashcroft | Solid-state physics[END_REF]. §2.2 is devoted to presenting a spectral algorithm to compute eigenvalues and eigenfunctions in the quasi-Bloch wave form. §3 presents some results for a simple quasi-periodic potential of the Mathieu type: the main emphasis is given to the appearance of "new bands" which look like being k-periodic, see Figs. 2 and3. In §4, we study eigenvalues and eigenfunctions for the equation [START_REF] Ashcroft | Solid-state physics[END_REF]. Again, when k is incommensurate, we observe k-periodic "new bands" which are seemingly parabolic, see especially Fig. 8. To these "new parabolic bands" correspond eigenstates which are extended, in sharp contrast to the critical states associated to the lower levels of Cantor-type nature as explained in §4.4. We shall even see that these critical states are localized on the iso-values of the potential. In §5, we study briefly what happens for different values of k.

Let us close this introduction emphasizing that defining a linearized potential in (1) like

V (t, x) def = cos(x) + ε sin(x) sin(kx -Ω(k)t), 0 ≤ ε 1, (2)
there are still periodicity properties that can be exploited:

V t + 2π/Ω(k), x = V (t, x), V t + 2kπ/Ω(k), x + 2π = V (t, x). (3)
This fact, already noticed in [START_REF] De Lange | Electrons in incommensurate crystals: spectrum and localization[END_REF], opens the way to perform a "quasi-Bloch decomposition" in 2-D for problems of the flavour of (1) for arbitrary values of k. The price to pay is of course a much higher computational cost as two indices will show up in the corresponding

Fourier series; but it also turns out that steady quasi-periodic problems ask for the same effort (see [START_REF] Surace | The Schrödinger equation with a quasi-periodic potential[END_REF] for an example and Appendix A for the corresponding illustration).

Remark 1

In every figure displaying a band structure, like Fig. 1, abscissas will always correspond to the Bloch "quasi-momentum" denoted hereafter and in (6) by κ. Ordinates stand for the associated energies E respectively.

Theoretical framework for "quasi-Bloch" decompositions

There are (at least) two ways to attack the eigen-functions problem given by a potential like (2): one follows techniques borrowed from existing papers on quasi-periodic equations, the other tries to take advantage of the space/time periodic structure (3) that still remains. We shall see in the sequel that both lead essentially (and fortunately) to the same algorithm in the two-dimensional case, but let us first recall some basic "quasi-periodic techniques".

2.1 Quasi-periodic potentials: a short overview of known results

Following [START_REF] Puig | Reducibility of linear equations with quasi-periodic coefficients: A survey[END_REF], let us begin with:

Definition 1 Let q(x) be a smooth function R → R; q is said to be quasi-periodic if there exist d ∈ N constants ω = (ω 1 , ..., ω d ) ∈ R d and a continuous function Q : R d → R 2πperiodic in each of its arguments such that:

∀x ∈ R, q(x) = Q( ω x) def = Q(ω 1 x, ..., ω d x).
The d constants ω are called the basic frequencies of q and Q is the (non unique) lift of q.

When dealing with quasi-periodic functions, we shall always use capital letters for the corresponding lifts. It is generally assumed that the components of ω are rationally independent otherwise the function q is simply periodic and Q might become ambiguous. Notice that this definition is compatible with the one we gave in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional periodic lattice. I. Homogeneous problems[END_REF]; it is a bit more general though.

It may sometimes be interesting to work with quasi-periodic functions endowed with an "initial phase" φ ∈ R d , for which q φ (x) = Q( ω x + φ). Roughly speaking, one can figure out what quasi-periodic functions are by thinking about functions of a single real variable admitting a multi-dimensional Fourier series: indeed, we write

q(x) = k∈Z d Q k exp(i < k, ω x >), Q k = 1 (2π) d T d Q(θ) exp(-i < k, θ >).dθ. (4)
The notation < ., . > stands for the scalar product in R d and the d-dimensional torus T d , for [0, 2π] d . It is generally assumed that the Fourier series (4) converges uniformly 3 . In this context, it makes sense to wonder if the elegant Floquet-Bloch theory for the onedimensional periodic 4 Schrödinger equation y + (E -q(x)) = 0, y(x) polynomially bounded [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF] extends in some sense to quasi-periodic potentials q. As soon as q is bounded, general spectral theory ensures that the corresponding operator is essentially self-adjoint on R; we shall denote H its self-adjoint extension to L 2 (R).

Proposition 1 ([36]) Let q φ (x) = Q( ω x + φ) where Q is continuous T d → R
and ω is rationally independent. Then the spectrum of the corresponding Schrödinger operator H on L 2 (R) doesn't depend on φ. Moreover, it doesn't have any isolated eigenvalues.

The study of quantum particles submitted an incommensurate potential started with [START_REF] Peierls | Quantum theory of solids[END_REF] and electrons moving inside a magnetic field. In a nutshell, the theory for quasi-periodic [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF] states that both eigenfunctions and eigenvalues E strongly depend on . For large , eigenfunctions are known to decay exponentially, this phenomenon is referred to as Anderson localization, see e.g. [START_REF] Fröhlich | Localization for a class of one-dimensional quasi-periodic Schrödinger operators[END_REF]. But for small and/or big E, there exist quasi-Bloch waves, that is to say eigenfunctions of the form (according to the former notation),

y(x) = exp(iκ(E)x)z(x), z(x) = Z( ω x), (6)
the lift Z being 2π-periodic in each of its arguments. KAM techniques are used in the proofs, which means that small divisors are to be controlled during the process. To do so, it is necessary to assume that ω is Diophantine, i.e. there exist constants C 0 , τ for which,

∀ ν = 0 ∈ Z d , | < ω, ν > | ≥ C 0 | ν| -τ , |κ(E)+ < ω, ν > | ≥ C 0 | ν| -τ . (7)
Quasi-Bloch waves for (5) with κ(E) = 1 2 < ω, t >, with t ∈ Z d and | < ω, t > | ≥ C 0 | t| -τ have been found in [START_REF] Dinaburg | The one-dimensional Schrödinger equation with quasi-periodic potential Funkt[END_REF][START_REF] Johnson | The rotation number for almost periodic potentials[END_REF][START_REF] Moser | An extension of a result by Dinaburg and Sinai on quasiperiodic potentials Comment[END_REF][START_REF] Eliasson | Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation[END_REF] along with the presence of gaps in the spectrum. Actually, to any quasi-periodic potential q and any eigenvalue E * can be associated a rotation number

β( , E * ) = (2π) -d lim x→∞ 1 x arg y (x) + iy(x) ,
where y solves (5) for the particular eigenvalue E * ; this number is independent of the selected solution y. It realizes a continuous function R 2 → R. Moreover, it is a nondecreasing function of E * for fixed. The Gap Labelling theorem expresses the fact that β( , E) is constant inside any connected component of the resolvent set:

Theorem 1 ( [START_REF] Johnson | The rotation number for almost periodic potentials[END_REF]) Let q be quasi-periodic with basic frequencies ω and I, an open interval inside a spectral gap. There exist integers k ∈ Z d such that:

∀E ∈ I, β( , E) = 1 2 < k, ω > .
The converse is also true: intervals of constancy for β are exactly the spectral gaps of H.

Definition 2 ([3]

) Let k ∈ Z d ; the resonance tongue of (5) associated to k is the set:

R( k) = ( , E) ∈ R 2 such that β( , E) = 1 2 < k, ω > .
This notion is interesting because it helps in validating numerical results indicating where spectral gaps can be for small ε, see [START_REF] Broer | Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena[END_REF][START_REF] Broer | Resonances tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation[END_REF]; we shall use it in Appendix A. Recently, an extension to the slightly more involved situation

y + (E -p(x) -q(x)) = 0, x ∈ R, (8)
where p is periodic has been treated in [START_REF] Gentile | Stability for the quasi-periodically perturbed Hill's equation[END_REF] (see also [START_REF] Broer | Resonances tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation[END_REF]); it led to the same kind of results, but for completeness we state the theorem:

Theorem 2 ([12]
) Let p : R → R be real analytic and periodic with frequency ω 0 ∈ R and such that the fundamental solutions of y + p(x) = 0 are quasi-periodic with a proper frequency Ω 0 ∈ R. Let furthermore q : R → R be real analytic and quasi-periodic with basic frequencies ω ∈ R d . Define ω := ( ω, ω 0 , Ω 0 ) ∈ R d+2 and assume ω is rationally independent with, moreover

∀ ν = 0 ∈ Z d+2 , | < ω, ν > | ≥ C 0 | ν| -τ ,
for two fixed positive constants C 0 and τ > d + 1 (Diophantine conditions). Then there exist 0 > 0 small and a Cantor set E ⊂ [-0 , 0 ] such that, for all ∈ E, the equation y + p(x) + q(x) = 0 admits a quasi-periodic solution of the form:

y(x) = exp(iΩ x) m,n∈Z d+1 ỹ m,n ( ) exp i(< m, ω > +nω 0 )x , (9)
and the above sum is absolutely and uniformly convergent for all x ∈ R and ∈ E.

Clearly ( 9) is a quasi-Bloch wave in the aforementioned sense. In the sequel, we shall mainly use Theorem 2 with d = 1. Let us finally stress that either (6) (and thus ( 9)) do agree with the ansatz suggested by experimentalists, see the conclusion of [START_REF] Rotenberg | Quasicrystalline valence bands in decagonal Al-Ni-Co[END_REF] and [START_REF] Torres | Quasiperiodic Bloch-like states in surface wave experiment[END_REF].

Remark 2 Theorem 2 can be recast into Eliasson's theory as we briefly explain now: any periodic Hill's equation is endowed with Floquet reducibility, meaning that a periodic change of variables takes it to constant coefficients. The existence of Bloch waves depends on the form of this constant coefficient Floquet matrix. Now, considering the quasi-periodic problem (8) and performing periodic reducibility, one obtains a small -quasi-periodic perturbation of a matrix with constant coefficients to which Eliasson's results can be applied.

An algorithm adapted to quasi-periodic potentials

The idea is to extend the spectral algorithm used in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional periodic lattice. I. Homogeneous problems[END_REF][START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -II. Impurities, confinement and Bloch oscillations[END_REF] to compute eigenvalues and Bloch waves in the periodic case to the more involved case of potentials q of the type (4). Indeed, assuming there exist quasi-Bloch waves solving the eigenvalue problem for H, we can project the differential equation onto the Fourier base corresponding to q. Taking into account for the quasi-periodic modulation z(x), we see that (6) rewrites

Ψ(x) = k∈Z d Ẑ k exp i < k, ω > +κ(E) x , (10)
where the coefficients of the lift Ẑ k are computed as in (4):

∀ k ∈ Z d , Ẑ k = 1 (2π) d T d Z(θ) exp(-i < k, θ >).dθ.
At this level, we follow [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional periodic lattice. I. Homogeneous problems[END_REF], §2.2 and compute for k, k ∈ Z d :

T ∂ xx {exp(i(< k, ω > +κ)x)} exp(-i(< k , ω > +κ)x).dx = -(< k, ω > +κ) 2 ,
where we used the L 2 -orthogonality of the base, and using ( 4)

T   ∈Z d Q exp(i < , ω x >)   exp(i < k -k , ω x >).dx = Q k-k .
The eigenvalue problem for H thus rewrites:

< k, ω > +κ 2 Ẑ k + k ∈Z d Q k-k Ẑ k = E(κ) Ẑ k . (11)
As long as k, k belong to a finite set, [START_REF] Fröhlich | Localization for a class of one-dimensional quasi-periodic Schrödinger operators[END_REF] leads to an eigenvalue problem for a (very big!) structured matrix when the eigenvectors Ẑ k are arranged in a convenient way. Truncations and error estimates can be computed following the method presented in Chap. 3, §11 of the book [START_REF] Taylor | Partial Differential Equations. I. Basic theory[END_REF], relying on the existence results of [START_REF] Broer | Resonances tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation[END_REF][START_REF] Gentile | Stability for the quasi-periodically perturbed Hill's equation[END_REF]. Concerning the convergence of the algorithm [START_REF] Fröhlich | Localization for a class of one-dimensional quasi-periodic Schrödinger operators[END_REF] as | k|, | k | → +∞, we recall from [START_REF] Magnus | Hill's equation[END_REF][START_REF] Zounes | Transition curves for the quasi-periodic Mathieu equation[END_REF] the following definition: Definition 3 Given any matrix A = (a i,j ) i,j with complex entries, its determinant is said to be of Hill's type if the following condition holds:

i,j |a i,j -δ i,j | < +∞, δ i,j the Kronecker symbol. ( 12
)
Dividing each line of ( 11) by its diagonal term is the first step to study whether the resulting matrix is of Hill's type, for instance when

Q is analytic, i.e. | Q k | ≤ C Q exp(-ξ| k|) or when Q is a finite sum of Fourier modes. Indeed, it comes Ẑ k + k ∈Z d Q k-k | < k, ω > +κ -E(κ)| 2 Ẑ k = 0,
to which we apply the Diophantine condition [START_REF] Dinaburg | The one-dimensional Schrödinger equation with quasi-periodic potential Funkt[END_REF] stating that:

∀ k ∈ Z d , 1 | < k, ω > +κ| ≤ C 0 | k| τ .
And the behavior for large Fourier modes with small divisors is ruled by terms of the type

| k| 2τ exp(-ξ| k -k |); other cases are like | < k, ω > +κ| -2 exp(-ξ| k -k |).
Hence if small divisors are very few, condition (12) can be met and according to the theory in [START_REF] Magnus | Hill's equation[END_REF], the diagonalization process [START_REF] Fröhlich | Localization for a class of one-dimensional quasi-periodic Schrödinger operators[END_REF] should converge (for the values of E(κ) for which the denominator doesn't vanish).

3 A steady quasi-periodic potential

Quasi-periodic perturbation of the Mathieu potential

The behavior of eigenvalues for the following time-independent equation has been studied theoretically in [START_REF] Fröhlich | Localization for a class of one-dimensional quasi-periodic Schrödinger operators[END_REF][START_REF] Surace | The Schrödinger equation with a quasi-periodic potential[END_REF][START_REF] Zheng | Renormalization procedure for the quasiperiodic Schrödinger equation[END_REF], numerically in [START_REF] Diener | Transition between extended and localized states in a one-dimensional incommensurate optical lattice[END_REF][START_REF] Hollingworth | Wave propagation in onedimensional optical quasiperiodic systems[END_REF][START_REF] Zounes | Transition curves for the quasi-periodic Mathieu equation[END_REF] and experimentally in [START_REF] Voit | Electronic structure of solids with competing periodic potentials[END_REF] (see also [START_REF] Torres | Quasiperiodic Bloch-like states in surface wave experiment[END_REF]):

1 2 ∂ xx Ψ + E -εq(x) Ψ = 0, q(x) = cos(x) + cos(kx). ( 13
)
From §2.1, we know that, provided the Diophantine condition (7) holds,

∀m, n = 0 ∈ Z 2 , |m + nk| ≥ C 0 n 2 ,
for ε small enough and/or E big, "quasi-Bloch waves" Ψ(x) = exp(iκx)z(κ, x) do solve the quasi-periodic eigenvalue problem [START_REF] Gérard | A mathematical approach to the effective Hamiltonian in perturbed periodic problems[END_REF]. In this last case, z(κ, .) and q share the same kind of quasi-periodicity: according to (4) they read,

q(x) = n,m Qn,m exp(i(n + mk)x), z(κ, x) = n,m
Ẑn,m (κ) exp(i(n + mk)x).

Of course, the potential is very well compressed in this base, Q±1,0 = 1 2 , Q0,±1 = 1 2 and the "quasi-momentum" κ(E) ∈ R doesn't belong to any Brillouin zone. This clearly allows for the use of the spectral algorithm [START_REF] Fröhlich | Localization for a class of one-dimensional quasi-periodic Schrödinger operators[END_REF] to compute the admissible values of E and their corresponding "quasi-Bloch waves" Ψ; we present numerical results in Appendix A. However, by analogy with the original problem (2), we decided to work with a slightly different equation: namely the corresponding potential reads with ε ∈ [0, 1],

q(x) = 1 2 cos(x) + ε cos(kx) , Q±1,0 = 1 4 , Q0,±1 = ε 4 . ( 15
)
This is rather a "small quasi-periodic perturbation" of a periodic problem hence we follow the ideas recalled in Prop. 2 and the algorithm proposed by Surace extends easily:

(n + mk + κ) 2 Ẑn,m + 1 2 Ẑn-1,m + Ẑn+1,m + ε Ẑn,m-1 + ε Ẑn,m+1 = 2E κ Ẑn,m . (16)
This special case of (11) loops on κ ∈ [-1, 1] (say!) and we chosed to restrict the computation to n, m ∈ {-15, 15}. The eigenvectors read ( Ẑ-15,-15 , Ẑ-15,-14 , ... Ẑ-15,15 , Ẑ-14,-15 , ...). The differential equation has therefore been reduced to the diagonalization of a 31 2 × 31 2 pentadiagonal symmetric matrix. We consider first two different cases k = 5 3 and k = 2 √ 5-1 : these two numbers are close to each other, but the second one is Diophantine thus enters completely in the theoretical framework of [START_REF] Gentile | Stability for the quasi-periodically perturbed Hill's equation[END_REF][START_REF] Surace | The Schrödinger equation with a quasi-periodic potential[END_REF] which rigorously ensures the existence of quasi-Bloch solutions.

3.2 Some numerical results with ε = 0.02 and ε = 0.2

The numerical outcome κ → E κ is presented in Figs. 2 and3 for ε = 0.02 and ε = 0.2 respectively. Several comments are in order: first, for k = 5/3 there's no need 5 for the algorithm [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -II. Impurities, confinement and Bloch oscillations[END_REF] as the period of the problem is 6π. Hence the corresponding Fourier base should be exp(inx/3), the Brillouin zone, κ ∈ [-1/6, 1/6] and the standard procedure explained in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional periodic lattice. I. Homogeneous problems[END_REF][START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -II. Impurities, confinement and Bloch oscillations[END_REF] can be used up to minor changes: its outcome is actually displayed on the left in Figs. 2 and3. Nevertheless, we also iterated the quasi-periodic algorithm [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -II. Impurities, confinement and Bloch oscillations[END_REF] in order to study both its accuracy and its transition between the commensurate and the incommensurate case. We observe that high in the spectrum, bands are always 1/3-periodic as expected; the lower ones should be also, but it isn't visible. However, something does show up inside the band gaps: a "new band" which doesn't look like being 1/3-periodic at all. It seems to survive the "band clustering" that destroys all the other ones when we put k = 2 √ 5-1 (on the right in Figs. 2 and3).

For ε = 0.2, Fig. 3 reveals the same type of pattern. That is, usual Bloch bands are easily noticeable on the left, and they are 1/3-periodic. There is an agreement between the left and middle figure, except for similar "new bands 6 ". Usual bands collapse when choosing k = 2 √ 5-1 because of the band nesting phenomenon illustrated in [START_REF] Hollingworth | Wave propagation in onedimensional optical quasiperiodic systems[END_REF] and become an intricate set that a computer cannot calculate properly. However, the new bands looking 5 Using it even seems to decrease a lot the accuracy of the numerical outcome. 6 which are spurious for k = 5/3, we explain why in §3.3.

Impurity bands ?

Concerning the presence of the "new bands" in case k = 5/3: we know from Floquet-Bloch theory [START_REF] Guillot | Semiclassical asymptotics in solid-state physics[END_REF] that since the problem is 6π-periodic, they shouldn't appear and this is what we get when using the algorithm based on exp(inx/3). Especially the problem is quite degenerate as many eigenvalues look like being very close to each other: this shows that the choice of the Fourier base by means of which one expresses both V (x) and Ψ(x) is of paramount importance in terms of numerical accuracy. For the same type of reason, putting ε = 0 inside (16) isn't likely to lead to satisfying numerical results as the selected Fourier base may not be the optimal one; hence the performances of the (quasi-periodic) algorithm on a periodic problem are quite low. Let us stress that in both cases (k rational and ε = 0), the algorithm ( 16) is built by projecting the differential equation ( 13) on the set of functions exp(i(n + mk)x); n, m in a finite subset of Z 2 . However, when ε = 0, the m index is useless; and when k = 5/3, we notice that for e.g. n = 3, m = 0 and n = 0, m = 5, L 2 -orthogonality is lost, along with the derivation of [START_REF] Fröhlich | Localization for a class of one-dimensional quasi-periodic Schrödinger operators[END_REF]. This last remark suggests that (16) should work correctly as long as orthogonality is kept within the selected finite set of indices (n, m). However, we ran the algorithm [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -II. Impurities, confinement and Bloch oscillations[END_REF] with another value of k = 1 2 (-1 + √ 5) in order to check its behavior in case the quasi-periodic perturbation is endowed with a frequency lower than the one of the underlying periodic potential. The numerical outcome is shown on Fig. 4; one notices at once the striking difference compared to Figs. 2 and3, namely the absence of any "new band". This suggests that the former results may not be spurious as ( 16) is able to produce neat band gaps in the quasi-periodic regime (31 Fourier modes have been used in this computation). One can also consult [START_REF] Voit | Electronic structure of solids with competing periodic potentials[END_REF][START_REF] Hafner | Electronic structure and stability of quasicrystals: quasiperiodic dispersion relations and pseudogaps[END_REF] where seemingly parabolic bands are being observed in a similar context. An heuristic explanation can be given relying on [START_REF] Nenciu | Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians[END_REF] where it is argued that band gaps in the spectrum are associated to long-range order in the potential, i.e. usually periodicity, see e.g. [START_REF] Korotyaev | Estimates on periodic potentials in terms of gap lengths[END_REF] for a rigorous illustration of this fact. However, even if one cannot expect to obtain forbidden energies for a completely disordered system, it is known that for small enough perturbations, gaps still appear in the spectrum. More precisely, we recall Theorem 3.1 from [START_REF] Nenciu | Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians[END_REF]:

Theorem 3 Let H 0 = -1 2 ∂ xx + V (x) with V periodic and H δ = -1 2 ∂ xx + V (x + δh(x)
) with h a C 3 (R) function having all its derivatives uniformly bounded by 1. Suppose [α, β] belong to the resolvent set of H 0 , then for δ small enough, there exist α ≤ α δ ≤ β δ ≤ β such that the same property holds for [α δ , β δ ] and H δ . Moreover,

lim δ→0 α δ = α, lim δ→0 β δ = β.
For δ > 0, periodicity is lost, but the result somewhat expresses the fact that a weaker short-range order still exists in H δ . Of course, long-range order is recovered in the limit δ → 0. In the present quasi-periodic context, we can think about using Theorem 3 with a linearized potential Q( ωx) := V (x) + δh(kx), for which the basic frequencies are ω = (1, k) (this situation is related to the one considered in [START_REF] Gérard | A mathematical approach to the effective Hamiltonian in perturbed periodic problems[END_REF], §IV.2). For k ≤ 1, the perturbation is sufficiently slowly-varying in order to maintain band gaps like in Fig. 4, but for k > 1, the result no longer applies and the spectrum's structure becomes more complex. Assuming that the "small perturbation" in εcos(kx) stands for the smoothered Coulomb interaction with impurities molten in the periodic lattice, one may be tempted to call the new bands impurity bands, following the terminology of [START_REF] Lax | One-Dimensional Impurity Bands[END_REF]. A possible interpretation of this phenomenon showing up for k ≥ 1 in this context is the overlapping of wave functions when more that one impurity appears in an elementary cell. These metastable states are of great use in real-life applications such as LASERs and fluorescent light emitters.

Time-dependent model in the incommensurate case

Owing to the preceding results, we won't try to explore the Cantor-like structure of the "collapsed bands"; since a Cantor set is infinite, but nowhere dense, it is impossible to get it computed accurately in practice (one may just say that something messy appears here or there inside the spectrum); we recall that a Cantor spectrum obtained from (8) has positive measure. Moreover, it isn't possible to label the bands in the absolute; only can be done to label the "new bands" among themselves.

2-D periodicity: reciprocal lattice and Bloch ansatz

We are now concerned with the study of the band structure for (1) with the linearized potential [START_REF] Benfatto | Electrons in a lattice with an incommensurate potential[END_REF]. Since it is endowed with some two-dimensional periodicity properties (3), it becomes completely possible to follow the canvas of the standard Bloch decomposition in order to handle the problem, whatever the values of k ∈ R -Q and ε > 0. Let us define the periodic lattice for V (t, x) in R 2 as spanned by the two vectors:

a 1 = (2πk/Ω(k), 2π) T , a 2 = (2π/Ω(k), 0) T .
The reciprocal lattice is spanned by b 1 and b 2 such that < a i , b j >= 2π.δ i,j , the Christoffel symbol. This yields:

b 1 = (0, 1) T , b 2 = (Ω(k), -k) T .
These vectors are useful to write down the Fourier series for the potential; indeed,

V (t, x) = m,n∈Z 2 Vm,n exp(imx) exp(in(kx -Ω(k)t)x), with V±1,0 = 1 2 , V1,±1 = ∓ ε 4 , V-1,±1 = ± ε 4 .
This produces a 2-D Brillouin zone (see [START_REF] Veerman | On Brillouin zones[END_REF] for some illustrations). However, we observe that there's no second derivative in time inside (1), thus the eigenvalue problem repeats itself exactly when t → t + 2π/Ω(k). Thus there is no quasi-periodic boundary condition in the time variable, and the Bloch ansatz reads simply

Ψ(κ, x) = exp(iκx)z κ (t, x),
the modulation z κ having the same periodicity as V , hence it can be written as a similar Fourier series. There is only one index κ which belongs to the interval

κ ∈ - max(1, |k|) 2 , max(1, |k|) 2 . ( 17
)
Following the last step in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional periodic lattice. I. Homogeneous problems[END_REF], we obtain an algorithm which is very close to the twodimensional version of ( 11): (we dropped the κ index for clarity here)

(m + nk + κ) 2 -2E ẑm,n + ẑm-1,n + ẑm+1,n + ε 2 ẑm-1,n+1 -ẑm-1,n-1 + ẑm+1,n-1 -ẑm+1,n+1 = 0. ( 18
)
Once again, it is possible to obtain a structured matrix on which iterative diagonalization routines can reveal their efficiency. It is because of the similarity between ( 11), ( 16) and ( 18) that we wrote at the beginning of §2 that both quasi-periodic and 2-D periodic approaches lead to the same algorithm. Remark however that presently, Diophantine assumptions on k shouldn't be needed in order to ensure the existence of Bloch wave solutions. On Fig. 5, we display the band structure obtained out of (18) for both n, m ∈ {-15, 15} (left) and n, m ∈ {-25, 25} (right; this one asked for several CPU-days to get done) with the value k = 2 √ 5-1 . This shows that involving many Fourier modes in the computation doesn't lead to qualitative changes in the solution, especially the "Cantor zones" stay in the same places. The nature of the "new bands" remain also roughly the same; the main difference is that the ones being closest to a "Cantor zones" seem to flatten. On Fig. 6, we present a zoom on the "new bands", especially in the vicinity of their crossing points. The lower ones cross only at the extremities of the interval [-k 2 , k 2 ] or in k = 0, but the higher ones cross elsewhere. Now, this last feature might be a consequence of a shortage of Fourier modes: when passing from n, m ∈ {-15, 15} to n, m ∈ {-25, 25}, some of the "new bands" seem to flatten (this is visible on the bottom of Fig. 6, somewhat less on the Even if we don't want to explore what lies inside the "Cantor zones", we display on Fig. 7 the band structures for two Fibonacci approximants of the golden mean k = 2 √ 5-1 computed by means of [START_REF] Gosse | Finite moment problems and applications to multiphase computations in geometric optics[END_REF] with n, m ∈ {-15, 15}. One observes that the upper part of the spectrum respects a periodicity in κ, even if it becomes more and more intricate. The amazing thing is that a parabolic structure seems to emerge in this area (as also suggested in [START_REF] Diener | Transition between extended and localized states in a one-dimensional incommensurate optical lattice[END_REF], Fig. 6).

The new bands of parabolic nature ?

It has been suggested in [START_REF] Diener | Transition between extended and localized states in a one-dimensional incommensurate optical lattice[END_REF] the possibility that an electron represented by a wave-packet whose motion is described by (1) may nearly not feel the quasi-periodic potential and then display a quadratic dispersion relation. The mere influence of the vibrating lattice would be noticeable only through an effective mass which would significantly differ from the one calculated out of the conduction band of the periodic lattice at rest. We follow this idea here and recall that the value m * 0.28 has been obtained for the static case in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional periodic lattice. I. Homogeneous problems[END_REF]. In Fig. 8, we juxtapose a parabolic approximation with the band picture where all the "Cantor states" have been removed. The agreement is quite interesting, but the values of the effective masses are notingly higher: we have m * 1.8 for the lowest states and m * 0.98 for the higher ones. The parabolic dispersion framework is appealing in terms of computational costs: indeed, it has already been explained in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional periodic lattice. I. Homogeneous problems[END_REF] that this allows for a direct implementation of shockcapturing routines to simulate the time-evolution of WKB wave-functions. However, the obstruction on this route can be formulated as follows: which modulation should be used to form the ansatz in case a band-crossing occurs ? (see e.g. [START_REF] Fermanian-Kammerer | An algorithm for quantum propagation through electron level crossings[END_REF])

Associated modulations: critical and extended states

In this section, we are about to recall some of the features observed in [START_REF] Kohmoto | Critical wave functions and Cantor-set spectrum of a one-dimensional quasicrystal model[END_REF][START_REF] De Lange | Electrons in incommensurate crystals: spectrum and localization[END_REF], that is to say, localized, critical and extended eigenstates. A state is said to be localized if it belongs to L 2 (R) and its position density decays exponentially when |x| → +∞. On the contrary, it is extended when its L 2 (R) norm is infinite. Critical states refer to those quasi-Bloch waves which position density exhibits spikes arranged in a periodic manner in the spacetime plane. Our algorithm ( 18) is unable to compute localized states, as it is meant to look for periodic modulations z κ (t, x) associated to some eigenvalue E κ . However, we can see on the top of Figs. 9 and 10 that critical states do show up; namely they are the ones associated with the two lowest "Cantor zones" in the band structure. In sharp contrast, the eigenstates associated to the "new parabolic bands" are extended, as can be seen on the bottom of Figs. 9 and 10. Moreover, one may observe that their position densities behave like the simpler ones in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional periodic lattice. I. Homogeneous problems[END_REF], Fig. 2: more precisely, the extended states corresponding to lower energies tend to be supported inside the potential well (even if it moves in time) whereas the higher ones are located above. The same seems to hold also for critical states, see Fig. 10 which displays spikes located on the potential's banks.

We found interesting to observe that critical eigenstates can actually be found at energy levels higher than extended ones. Indeed, to the critical state on top of Fig. 10 corresponds a "Cantor band" higher in the structure than the parabola associated to the extended state in the bottom of Fig. 9. This hasn't been noticed in [START_REF] De Lange | Electrons in incommensurate crystals: spectrum and localization[END_REF].

On Fig. 11, we display the iso-lines of position densities associated to certain eigenstates with ε = 0.15: the left one refers to the lowest extended state, which is mainly situated in the vincinity of the potential's minima. The middle one is related to the highest critical state in the second "Cantor zone": one can notice how it follows the potential's high iso-lines. The right one corresponds to the extended state located immediately above in the spectrum; it is mainly located outside the potential's minima too, but its structure is obviously very different as it may be a good candidate to figure a conduction electron. They all correspond to the 3-D representations of Figs. 9 and 10. Higher eigenstates reveal more uniformly located oscillations. We didn't observe any localized states like in [START_REF] De Lange | Electrons in incommensurate crystals: spectrum and localization[END_REF]; however, their presence in the spectrum could be questionable as the potential ( 2) is genuinely 2-D periodic. Hence its eigenstates should be given by 2-D Bloch waves having only the particularity of being exactly periodic in time. However, the position densities of the critical states we found by means of (18) follow nicely the iso-lines of the potential (2). We shall see in the next section that this is quite a robust property since it still holds while changing the value of the parameter ε which controls the strength of the modulation: a stronger modulation giving a more oscillating potential. Increasing the value of the parameter ε doesn't seem to change the structure of the overall band picture: namely, it widens the "Cantor-like zones" inside the spectrum, as can be seen on Fig. 12 with n, m ∈ {-15, 15}. The "new bands" aren't much changed during the process, except that their interval of variation has shrinked. One can imagine that for a really big value of ε, no band can show up no more, and the spectrum will therefore display a completely singular structure, like on the top of Fig. 7 (or in [START_REF] De Lange | Electrons in incommensurate crystals: spectrum and localization[END_REF], Fig. 1). The associated eigenfunctions would display the Anderson localization.

0.9 0.9 0 0 -0.7 -0.7 -0.9 -0.9 For completeness, we also displayed the behavior of the (critical) ground state when ε is increased on Fig. 13. The main observation is that the spikes keep on being located on the iso-values of the potential, even when it shows quite severe oscillations in the space/time plane. This suggests that the whole algorithm [START_REF] Gosse | Finite moment problems and applications to multiphase computations in geometric optics[END_REF] produces solutions which remain consistent with the data, even if they reveal a more singular structure than e.g. the Bloch waves in [START_REF] Gosse | Multiphase semiclassical approximation of the one-dimensional harmonic crystal. I. The periodic case[END_REF].

5 What happens for different values of k ?

We have seen in the last two sections that for both static and time-dependent models, "new bands" appear and look like being k-periodic. Hence it might be that reducing the value of k is likely to make them more difficult to observe, like in §3.3 (however, the interpretation as impurity bands doesn't really hold here). This is what we want to study in this section working with the inverse k = √ 5-1 2 . We shall also have a look at k = π 3 , for which the "parabolic structure" previously discussed should appear.

k is smaller than one

We still consider the potential (2) and we use [START_REF] Gosse | Finite moment problems and applications to multiphase computations in geometric optics[END_REF] to investigate numerically its eigenvalues in case k = "New bands" might really be a quasi-periodic phenomenon which appearance stems only from the set of basic frequencies present in the potential V . In both cases given by ( 2) and ( 15), basic frequencies are ω = (1, k) at any fixed time t; "new bands" could develop better with |k| getting bigger, meaning that the (small) perturbation term oscillates at a frequency higher than the one of the lattice at rest. This gives for both potentials a threshold value of k 1, in agreement with the numerical results. On the contrary, owing again to Figs. 4 and 14, the number of band gaps seems to increase when ε/|k| grows.

We also present position densities for some eigenstates chosen around some energy level like in Fig. 11; results are shown in Fig. 15. One sees that as ε grows, these states pass from being "nearly extended" (for ε = 0.05) to clearly critical (for ε ≥ 0. for ε = 0.05 (left), ε = 0.15 (middle) and ε = 0.5 (right).

k is bigger than one

In order to decide whether the (incommensurate) value of k has an influence on the "new bands", we ran the algorithm [START_REF] Gosse | Finite moment problems and applications to multiphase computations in geometric optics[END_REF] with k = π 3 ≥ 1 and ε = 0.15. Looking at Fig. 16, one can see the big difference compared to Fig. 14: no band gaps remain, and the upper part of the spectrum doesn't really appear like a Cantor set anymore. Seemingly "parabolic new bands" are visible and display a k-period as in the previous sections; there is some sort of similarity with Fig.

Conclusion

In order to compute accurately the spectrum of a Schrödinger equation modelling a onedimensional modulated (or deformed) incommensurate crystal without any tight-binding approximation, we derived in this paper a spectral algorithm which can be built out of the recent theory of quasi-periodic functions or by taking advantage of periodicity properties of the potential in a 2-D space. The numerical results revealed several unexpected features:

• New "seemingly parabolic bands" emerge in the spectrum. The band diagram is therefore split into "Cantor-like" domains which result from the band nesting phenomenon, [START_REF] Hollingworth | Wave propagation in onedimensional optical quasiperiodic systems[END_REF] and these "impurity bands" which turn out to be k-periodic. When computing the corresponding effective masses, one finds values much bigger than the ones coming from the usual conduction band of the unperturbed lattice (in our case, the Mathieu equation treated in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional periodic lattice. I. Homogeneous problems[END_REF][START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -II. Impurities, confinement and Bloch oscillations[END_REF][START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -III. From ab-initio models to WKB for Schrödinger-Poisson[END_REF]). Hence two competing effects appear: metallic conductivity can be achieved this way, like e.g. in highly-doped semiconductors, but scattering effects due to quasi-periodicity are important. • Two different types of eigenstates appear: in the terminology of [START_REF] Kohmoto | Critical wave functions and Cantor-set spectrum of a one-dimensional quasicrystal model[END_REF][START_REF] De Lange | Electrons in incommensurate crystals: spectrum and localization[END_REF], we speak about critical states generally found in the bottom of the spectrum and always associated with a Cantor-like area. These states are very peaked and their spikes are located on the isolines of the potential. The other type of eigenstate is the more usual extended state, which looks essentially like a Bloch wave: those are the only ones which may represent conduction electrons. • The value of the wave number k in ( 2) is very important in many aspects: even if one assumes it is Diophantine, its magnitude has a huge influence on the overall spectrum picture, recall e.g. the Figs 14 and 16: the threshold value in this case looks like being k 1, which means that "new bands" show up only in case the frequency of the small perturbation is higher than the one of the underlying periodic lattice. This seems to be supported by Theorem 3.1 in [START_REF] Nenciu | Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians[END_REF], which holds in multi-dimension and states essentially that weak and slowly-varying deformations of periodic potentials still admit band gaps. This suggests that a semiclassical WKB approximation to (1) should take into account for these features. The main obstruction in this direction lies in understanding what happens at the band-crossing points, since it looks clear from Fig. 7 and 16 that the parabolas extend very far inside the band diagram on the top of the spectrum.

A Numerical visualization of a continuous "almost-Mathieu" model

For completeness, and even if this case doesn't exactly fit in our general framework 7 , we display here the numerical outcome of the algorithm written in [START_REF] Surace | The Schrödinger equation with a quasi-periodic potential[END_REF] to solve the problem [START_REF] Gérard | A mathematical approach to the effective Hamiltonian in perturbed periodic problems[END_REF]; it is just a special case of [START_REF] Fröhlich | Localization for a class of one-dimensional quasi-periodic Schrödinger operators[END_REF]. Namely what we want to check is the ability of this numerical processing to recover well-known results, like e.g. the existence of band gaps for ε small enough. Unfortunately, it isn't possible to achieve conclusive computations for ε ≤ 0.1 with a reasonable number of Fourier modes (typically, between 31 and 51) because the corresponding outcome is blurred with spurious oscillations (like on the left of Fig. A.1). In this figure, one sees that for ε = 0.5, a lot of eigen-states are constant in κ but, for all values of ε, at least one band gap shows up in the structure. For ε > 0.7, the algorithm becomes unstable; we recall that in this case, the parameter ε acts on the whole potential, which isn't really a perturbated periodic function no more.

We finally wanted to observe what happens for a small value of k in this case, and we took again k = √ 5-1 2 ; results are displayed in Fig. A.2. This seems to have no effect on the presence of "new bands", whose presence can be questionable as the lower ones on Figs.

A.1 and A.2 appear inside a range of energies where the computations reported in [START_REF] Broer | Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena[END_REF][START_REF] Zounes | Transition curves for the quasi-periodic Mathieu equation[END_REF] predict a band gap. They don't seem to be neither parabolic nor k-periodic as in the former sections, hence we think they are spurious. Actually, our outcome with ε = 0.5 could also be compared with Fig. 3 in [START_REF] Diener | Transition between extended and localized states in a one-dimensional incommensurate optical lattice[END_REF] where periodic approximations are studied by means of a spectral scheme like in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional periodic lattice. I. Homogeneous problems[END_REF]; the corresponding potential is displayed on Fig. 2 in the same 7 It is better suited for 1-D quasicrystals. article. Similarly, something appears in the band gaps for another case study of the type [START_REF] Gérard | A mathematical approach to the effective Hamiltonian in perturbed periodic problems[END_REF] reported in [START_REF] Hollingworth | Wave propagation in onedimensional optical quasiperiodic systems[END_REF]: see Fig. 3, p/q 30/61 and p/q 31/61 In order to provide some explanation about the figures A.1 and A.2, we turn back to the notion of "resonance tongue", see Definition 2 and [START_REF] Broer | Resonances tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation[END_REF][START_REF] Broer | Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena[END_REF]. In case the eigenvalue problem is given by ( 13), we observe that ω = (1, k) ∈ R 2 thus the set of resonances is given by As the frequency vector ω is identical, we may think that the little gap around E = 0.4 on both figures 2 and 3 has the same origin. A simpler resonance is proposed in Fig. 2 of [START_REF] Broer | Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena[END_REF] (according to their notation, E = a 2 2 and ε = b 2 ) for ( 13): it is based on the choice i = 0, j = 1 in (A.1) and yields E 0.33. It marches our results quite well (and also Fig. 5.1 in [START_REF] Zounes | Transition curves for the quasi-periodic Mathieu equation[END_REF]) but one may prefer our former value because it meets also the outcome of §3. Still, the basic frequencies vector ω is the same compared to the one which led to the results of Fig. 4 even if the potential is different; nevertheless, we observe that a gap opens around the same energy value in that case. Looking at Figs. 2.1, 3.1, 5.1 and 5.2 in [START_REF] Zounes | Transition curves for the quasi-periodic Mathieu equation[END_REF] while according to their notation (ε = ε/2 and E = δ/2), we see that very different algorithms led to the finding of a very stable gap (in terms of k) for this range of energies. Of course, this is too weak as a validation for our results. However, the fact that our gaps are stable for small values of ε and are roughly located around commonly found resonances is rather encouraging.

Remark 3

The results in this appendix are related to the ones in [START_REF] Broer | Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena[END_REF] (and also in [START_REF] Zounes | Transition curves for the quasi-periodic Mathieu equation[END_REF]); reference [START_REF] Broer | Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena[END_REF] concentrates on the eigen-problem (5) where q(x) = cos(ω 1 x) + cos(ω 2 x) with incompatible frequencies. Different notations are used as their (a 2 , b) correspond here to (E, ). [START_REF] Broer | Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena[END_REF]. Later, a study of instability pockets is conducted for q(x) = cos(ω 1 x) + cos(ω 2 x) + cos((ω 1 + ω 2 )x), which isn't touched here. Similarly as [START_REF] Zounes | Transition curves for the quasi-periodic Mathieu equation[END_REF] Figure 3.1, the authors of [START_REF] Broer | Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena[END_REF] study numerically the Lyapunov exponent λ and the rotation number ρ associated to [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF], both as functions of E in Figs. 4567. At last, the remaining of [START_REF] Broer | Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena[END_REF] is concerned with a very detailed numerical study of phenomena near the collapse line of resonances, thus goes beyond the scope of the present work.

Fig. 1 .

 1 Fig. 1. Band structures for ε = 0.15 with commensurate (middle) and incommensurate models: k = √ 5-1 2

- 1 .Fig. 2 . 2 √ 5 - 1 (Fig. 3 . 2 √ 5 - 1

 122513251 Fig. 2. Band structure with ε = 0.02 and k = 5/3 (left and middle), k = 2 √ 5-1(right). The figure on the left is obtained with the optimal Fourier base exp(inx/3).

Fig. 4 . 5 2

 45 Fig. 4. Spectrum with k = -1+ √ 5 2 and ε = 0.02 (left), ε = 0.05 (middle), ε = 0.2 (right).

4. 2 Fig. 5 .

 25 Fig. 5. Band structures with n, m ∈ {-15, 15} (left) and n, m ∈ {-25, 25} (right).

Fig. 6 .Fig. 7 .

 67 Fig. 6. Band crossings with n, m ∈ {-15, 15} (left) and n, m ∈ {-25, 25} (right).top). It might therefore be that, for a even higher set of Fourier modes, only remain pieces of the k-periodic parabolas displayed on the left of Fig.8separated by Cantor-like zones.

Fig. 8 .

 8 Fig. 8. Band structure for the incommensurate model: parabolic approximations periodically repeated (left), real values coming from the spectral algorithm (right) and superimposed (middle).

Fig. 9 .

 9 Fig. 9. Position density for increasingly energetic modulations with κ = 0: ground state (top) and extended one corresponding to the lower parabola (bottom).

Fig. 10 .

 10 Fig.10. Position density for increasingly energetic modulations with κ = 0, continued: critical state in the intermediate Cantor set (top) and extended one corresponding to the higher parabola (bottom).

Fig. 11 .

 11 Fig. 11. Lower extended (left), critical (middle) and higher extended (right) eigenstates. (absissae are for space, ordinates for time)

4. 5 5 -Fig. 12 .

 5512 Fig. 12. Band structures for ε = 0.05 (left), ε = 0.15 (middle) and ε = 0.5 (right).

Fig. 13 .

 13 Fig.[START_REF] Gérard | A mathematical approach to the effective Hamiltonian in perturbed periodic problems[END_REF]. Ground states superimposed with potential's iso-values for ε = 0.05 (left), ε = 0.15 (middle) and ε = 0.5 (right).

√ 5-1 2 ≤ 1 :Fig. 14 .

 2114 Fig. 14. Spectrum for ε = 0.05 (left), ε = 0.15 (middle) and ε = 0.5 (right) with k = √ 5-1 2 .

Fig. 15 .

 15 Fig. 15. Eigenstates with k = √ 5-1 2

5 -Fig. 16 .

 516 Fig. 16. Spectrum for k = π 3 , ε = 0.15: full picture (left) and detail (middle); comparison with k = π 4 (right).

20

 20

Fig. A. 1 . 2 √ 5 - 1 :

 1251 Fig. A.1. Spectrum for Surace's model with k = 2 √ 5-1 : inconclusive ε = 0.1 (left), ε = 0.3 (middle) and ε = 0.5 (right).

- 1 .Fig. A. 2 .√ 5 - 1 2

 1251 Fig. A.2. Spectrum for Surace's model with k = √ 5-1 2 : ε = 0.05 (left), ε = 0.15 (middle) and ε = 0.5 (right).

  jk), (i, j) ∈ Z 2 , (A.1) hence is dense in R. Owing to the continuity of the rotation number β, we just want to check that the gaps we found in the preceding figures for ε small do correspond to resonances of the form (A.1) for small values of i, j. Indeed on Fig. A.1, for both ε = 0.1 and ε = 0.3, we observe a gap roughly in [0.25,0.45]; this may emanate from the elementary resonance:

2 .

 2 Concerning Fig. A.2, there seems to be a gap inside the interval [0.1,0.15] for both ε = 0.05 and ε = 0.15; this may correspond to the most elementary resonance:

  Figures 1 and 2 in [4] are a different view of the band structure disregarding the κ-dependence of E(κ, ) but emphasizing the influence of the size of . Summarizing, in the present paper, we plot κ → E for different values of (see Figs. A.1 and A.2) whereas → E is shown in Figs. 1 and 2 of

A precise definition of this term is to be given in §2.1 of this text.

for Kolmogorov, Arnold and Moser.

This trongly depends on the smoothness of Q.

This is the case where d = 1.


Laurent Gosse
email: l.gosse@ba.iac.cnr.it

Multiphase semiclassical approximation of the one-dimensional harmonic crystal ‡

Keywords: AMS classification scheme numbers: Primary 81Q05, 81Q20; Secondary 35L65, 65M06

One-dimensional electronic conduction is investigated in a special case usually referred to as the harmonic crystal, meaning essentially that atoms are assumed to move like coupled harmonic oscillators within the Born-Oppenheimer approximation. We recall their dispersion relation and derive a WKB system approximately satisfied by any electron's wave-function inside a given energy band. This is then numerically solved according to the theory of K-branch solutions. Numerical results are presented in case atoms move with one-or twomodes vibrations; at last, we include the case where the Poisson self-interaction potential influences the electrons dynamics too.

Introduction

Preliminaries

It is a familiar fact that electrons are able to move over long distances inside certain materials; this phenomenon manifests itself, for instance, through the high electrical conductivity of metals. However, even in purest metals, electrons are influenced by the underlying lattice made of the atoms consituting the crystal itself. A very reasonable assumption is to consider the repartition of these atoms as periodic; such an idealization allows to take advantage of Bloch's theory which explains how valence/conduction electrons can move rather freely and have a well-defined (Bloch) momentum. Indeed, Bloch's theorem is one example of coherent electronic transport: in a perfect crystal at zero temperature, there is a great superposition of phasecoherent waves with similar wavenumbers which results in a frictionless flow inside the material (at low densities, Pauli's exclusion principle doesn't have very sensible effects). These electrons move according to the energy bands thus can survive multiple scattering events with the atoms without getting localized and accumulating somewhere. Unfortunately, finite conductivity is observed in practice because no sample will be pure enough to be a perfectly periodic lattice, and even if this idealization was realizable, vibrations due to thermal excitation would perturb the picture. The effects of these vibrations constitute the main point we want to study in this paper within a simplified framework.

We are interested in deriving approximate WKB solutions of certain Schrödinger equations in one space dimension, where short-range scattering effects other than those considered in the Bloch decomposition will be neglected. In other words, we shall work on ballistic transport of electrons for which the most important long-range interaction is the Coulomb force. In case one assumes translational invariance in two directions, three dimensional objects can be modelled to some extent by means of a one-dimesional mean-field equation. We shall follow these ideas which have already been studied in former works, [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a onedimensional crystalline lattice -I. Homogeneous problems[END_REF][START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -II. Impurities, confinement and Bloch oscillations[END_REF][START_REF] Gosse | Multiphase semiclassical approximation of an electron in a onedimensional crystalline lattice -III. From ab-initio models to WKB for Schrödinger-Poisson[END_REF]. However, we go now one step further considering that atoms vibrate as "coupled harmonic oscillators". The Hamiltonian for one electron reads (see [START_REF] Ashcroft | Solid-state physics[END_REF], page 430), with obvious notation,

H f ull = p 2 2m + α∈Z P 2 α 2M + α∈Z V ions (|X α -X α-1 |) + α∈Z V ion-e -(|x -X α |). (1)
Parameters (p, x, m) refer to some electron and (P α , X α , M ) α to the collection of atomic cores. One could also consider a cloud of electrons indexed by β ∈ Z and the first term in (1) should be replaced by

β∈Z p 2 β 2m + 1 2 β =β e 2 |x β -x β | .
Then Coulomb self-interactions may be treated via "mean-field", which would lead to some Hartree equation, cf. [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a onedimensional crystalline lattice -III. From ab-initio models to WKB for Schrödinger-Poisson[END_REF]; this will be briefly considered in the numerical results of §4.3. But first, let us concentrate on the one-particle model [START_REF] Ashcroft | Solid-state physics[END_REF]. We shall assume the following hypotheses to hold throughout the whole paper:

• Ionic cores are coupled oscillators, i.e. V ions (|X α -X α-1 |) = 1 2 M ω 2 (X α -X α-1 ) 2 .
• They are treated classically (Born-Oppenheimer approximation) because m M 1.

• They influence electrons but not the opposite (there are no polarons, cf. [START_REF] Ashcroft | Solid-state physics[END_REF][START_REF] Nieto | A generalized mean-field approach to the polaron[END_REF]).

Modelling of ionic cores vibrations

Without V ions , ions would arrange periodically Xα = 2πα, α ∈ Z, like in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a onedimensional crystalline lattice -I. Homogeneous problems[END_REF]; now, let's call U α (t) := X α (t) -2πα the displacement around these abscissae "at rest". The cores' equation is decoupled thanks to our third hypothesis,

d dt 2 U α (t) -ω 2 U α+1 (t) -2U α (t) + U α-1 (t) = 0.
As for the continuous wave equation, it can be solved by Fourier series:

Q(t, k) = α∈Z U α (t) exp(-ik2πα) ⇒ d dt 2 Q(t, k) = Ω(k) 2 Q(t, k).
The dispersion relation reads, (see also [START_REF] Ashcroft | Solid-state physics[END_REF])

Ω(k) = 2ω| sin(kπ)|, k ∈ B := - 1 2 , 1 2 ,
with B the Brillouin zone associated to the Bravais lattice for the ionic cores "at rest". U α (t) can be written as a continuous summation of plane waves:

U α (t) = 1 2 0 A(k) exp i(k2πα -Ω(k)t) + B(k) exp -i(k2πα -Ω(k)t) .dk. ( 2
)
The Schrödinger equation for the electron's wave function ψ becomes accordingly,

i ∂ t ψ + 2 2m ∂ xx ψ = V per x + U α (εt) ψ, (3)
with V per (x) α V ion-e -(|x -Xα |) can be considered 2π-periodic and smooth; the dimensionless paramenter ε = Te -/ Tions is defined as the ratio of characteristic times for the movements of ions and electrons, respectively. This modelling has already been incontered in several papers; let us quote [START_REF] Niu | Quantum adiabatic particle transport[END_REF][START_REF] Sundaram | Wave-packet dynamics in slowly perturbed crystals: gradient corrections and Berry-phase effects[END_REF] where it is referred to as "the deformed crystal". Moreover, many studies have been made in the field of "incommensurate crystals", see [START_REF] De Lange | Electrons in incommensurate crystals: spectrum and localization[END_REF][START_REF] Quilichini | Phonon excitations in quasicrystals[END_REF][START_REF] Mishra | One-dimensional photonic crystal: the Kronig-Penney model[END_REF], emphasizing the so-called modulated Kronig-Penney model. This is somewhat an extreme case for which the potential showing up in (3) isn't periodic any more thus the Brillouin zone of the resulting crystal is reduced to a point. We believe that new methods are needed for tackling such a problem of non-periodic homogenization which goes beyond the scope of the present paper. Indeed all this text deals only with perturbed potentials (as showing up in the right-hand side of (3)) endowed with some perdiodicity.

WKB asymptotic expansions of wave functions

We assume the reader to be familiar with Bloch's theory; a comprehensive presentation is given in [START_REF] Ashcroft | Solid-state physics[END_REF], see also [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF][START_REF] Dimassi | Semiclassical asymptotics in magnetic Bloch bands[END_REF][START_REF] Guillot | Semiclassical asymptotics in solid-state physics[END_REF][START_REF] Hövermann | Semiclassical limit for the Schrödinger equation for a short scale periodic potential[END_REF][START_REF] Gosse | Multiphase semiclassical approximation of an electron in a onedimensional crystalline lattice -I. Homogeneous problems[END_REF]. In this section, we shall show that for simple enough displacements (2) (typically with few vibration modes), it is possible to derive a WKB approximation for the wave function ψ solution of (3) for smooth V per . For simpliciy, we switch at once to "atomic units" for which = m = e = 1 and seek a convenient Ansatz as variables are rescaled according to (t, x) → (εt, εx) with 0 < ε 1; ε = 0 would mean that phonon scattering acts instantaneously.

Bloch theory and the Eikonal equation

We are now basically interested in the following equation:

iε∂ t ψ + ε 2 2 ∂ xx ψ = Ṽper t, x ε ψ, Ṽper (t, y) = V per (y + u(t, y)), (4)
where y = x/ε is the microscopic variable, Ṽper is supposed to be C ∞ in both variables, the motion u is smooth and periodic in both its variables, and V per is 2π-periodic. The simplest case where u is only endowed with a unique vibration mode is displayed in Fig. 1. Let us assume that for all t, Ṽper (t, y + 2π/k) = V per (t, y); thus one can consider a slightly generalized Bloch eigenvalue problem:

∀t ∈ R, - 1 2 ∂ yy Ψ n,κ (t, y) + Ṽper t, x ε Ψ n,κ (t, y) = E n (t, κ)Ψ n,κ (t, y), (5)
for any t ∈ R + and Ψ n,κ (t, y) = exp(iκy)z n,κ (t, y) a Bloch state. We observe that we only need y ∈]0, 2π/k[ and the Brillouin zone corresponding to Ṽper is smaller than B,

κ ∈ B = - k 2 , k 2 , |k| ≤ 1 2 .
Since there's no differentiation in time inside [START_REF] Brenier | Averaged multivalued solutions for scalar conservation laws[END_REF], standard results remain valid (see e.g. [START_REF] Markowich | A Wigner-function approach to semiclassical limits: electrons in a periodic potential[END_REF]) and ensure that for any fixed t and κ ∈ B, there exists a complete set of eigenfunctions Ψ n,κ (t, .) ∈ L 2 (0, 2π/k) with countably many eigenvalues

E 1 (t, κ) < E 2 (t, κ) < • • • < E n-1 (t, κ) < E n (t, κ) < • • • depending smoothly on time.
The set {E n (t, κ); κ ∈ B} is called the n th energy band whereas Ψ n,κ is a n th Bloch state having the form exp(iκy)z n,κ (t, y) for a certain 2π/k-periodic modulation. Modulations can be normalized so as to form an orthonormal base of L 2 (0, 2π/k) for each t > 0.

At this level, we introduce a WKB Ansatz for ψ considering a two-scale amplitude:

A t, x, y = x ε = A 0 (t, x, y) + εA 1 (t, x, y) + • • • ; A t, x, y + 2π/k = A(t, x, y). ( 6
)
Plugging the approximation A(t, x, x/ε) exp(iϕ(t, x)/ε) inside ( 4) and balancing the O(1) terms following carefully [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF][START_REF] Dimassi | Semiclassical asymptotics in magnetic Bloch bands[END_REF][START_REF] Guillot | Semiclassical asymptotics in solid-state physics[END_REF] can be shown to lead to an Hamilton-Jacobi equation for the phase:

∂ t ϕ + E n (t, ∂ x ϕ) = 0. (7)

Derivation of the transport equation

The principal amplitude stems from balancing O(ε) terms and is usually handled by means of the "Feschbach method", see the paper [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF] to which we refer for a detailed presentation of the computation. Here we shall limit ourselves to specify the changes occuring in the derivation because of the time-dependence of Ṽper . Actually, everything proceeds as in the Appendix A of [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF], except for the last step that we explain now §: let us denote L the usual geometric optics transport operator associated to the Bloch theory:

La := E n (t, ∂ x ϕ)∂ x a + 1 2 ∂ x (E n (t, ∂ x ϕ))a,
where E n stands for the partial derivative with respect to κ. The principal amplitude is defined as a 0 (t, x) such that there holds: A 0 (t, x, y) = a 0 (t, x)z n,κ (t, y) for κ = ∂ x ϕ(t, x). From (A.4), (A.5), (A.6) in [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF], we obtain the following equation for a 0 :

∂ t a 0 + a 0 2π/k 0 g n ∂ t g n .dy + E n (t, ∂ x ϕ)a 0 2π/k 0 g n ∂ x g n .dy + La 0 = 0, ( 8
)
where g n (t, x, y) = z n,κ=∂xϕ(t,x) (t, y). Now, as a consequence of the modulation's normalization, z n,κ (t, .) L 2 (0,2π/k) = 1, we observe that the first coefficient acting on a 0 is purely imaginary,

∂ t 2π/k 0 g n g n .dy = 0 = 2 2π/k 0 g n ∂ t g n .dy ,
and since E n ∈ R, the next one as well. It remains to make use of the definition of g n , the chain rule, and the equation ( 7) to compute:

2π/k 0 g n ∂ t g n + E n (t, ∂ x ϕ)∂ x g n .dy = 2π/k 0 z n,∂xϕ(t,x) ∂ t z n,κ κ=∂xϕ(t,x) .dy.
We denote this term β(t, x) ∈ iR; it identifies with a Berry phase, [START_REF] Berry | Proc. R. Soc. Lond. A[END_REF], and stems from the time-dependence of the Bloch states. In case another slowly-varying potential V (t, x)ψ was added on the right-hand side of (4), one would get another phase-shift too reading

∂ x V (t, x) 2π/k 0 z n,∂xϕ(t,x) ∂ κ z n,κ κ=∂xϕ(t,x)
.dy ∈ iR like in [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF][START_REF] Dimassi | Semiclassical asymptotics in magnetic Bloch bands[END_REF][START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -II. Impurities, confinement and Bloch oscillations[END_REF]. Anyway, it now remains to multiply the resulting equation ( 8) by ā0 and take its real part to derive the usual continuity equation for |a 0 | 2 :

∂ t |a 0 | 2 + ∂ x E n (t, ∂ x ϕ)|a 0 | 2 = 0. ( 9
)
All in all, we have shown that an approximate n th -band solution for (4) reads

ψ ε (t, x) = a 0 (t, x) exp iϕ(t, x)/ε z n,∂xϕ(t,x) t, x/ε , t ≥ 0, (10)
§ We shall use the same notations for easiness in reading.

and evolves in time according to the WKB system made of ( 7) and [START_REF] Gosse | Using K-branch entropy solutions for multivalued geometric optics computations[END_REF]. A first consequence is that an initial datum concentrated on a given band will necessary give rise to an approximate solution in the same band. Lattice vibrations aren't supposed to make electrons trigger interband transitions; hence from now on, we consider some electron moving according to some energy band whose index n ∈ N is fixed.

"Lattice tracking" phenomenon

Clearly, each primitive cell of the "perturbed lattice" contains 1 |k| ∈ N atomic cores. We modified the algorithm in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -II. Impurities, confinement and Bloch oscillations[END_REF], §2.2 in order to compute numerically E n (t, κ), Ψ n,κ (t, y) by means of a spectral method. We write down:

Ψ n,κ (t, y) = ∈Z Ψ n,κ (t) exp(i(n + κ)y/k), Ψ n,κ (t) = 2π/k 0 z n,κ (t, y) exp(-iny/k) 2π .dy.
The same decomposition is carried out for the perturbed potential and one is led to diagonalize a matrix H κ similar to that in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -II. Impurities, confinement and Bloch oscillations[END_REF] with kinetic energy terms reading now

k 2
2 (κ + ) 2 , = -N, ..., N . Modulations z n,κ are recovered by Fast Fourier transform (fft) of corresponding eigenvectors and there holds for some t ∈ R + , n ∈ N: Since there are 1/k atoms per period, bands "stick together" and constitute sets of 1/k (for k small, there would be "bands of bands" emerging inside a tiny Brillouin zone). In the ground state, each of the 1/k modulations is very peaked around the corresponding atom in the elementary cell: see Fig. 2 where Ṽper (t, y) = cos y + π 2 sin(ky + Ω(k)t) . It is interesting to make a link with a feature pointed out in [START_REF] Sundaram | Wave-packet dynamics in slowly perturbed crystals: gradient corrections and Berry-phase effects[END_REF], namely the tendency for the lattice to drag the electron with its displacement motion, called "lattice tracking". First, observe that even if supplying the WKB system ( 7)- [START_REF] Gosse | Using K-branch entropy solutions for multivalued geometric optics computations[END_REF] with initial data such that a 0 ≡ 1 and ∂ x ϕ ≡ 0, we would have ψ ε (t, x) = ψ ε (t = 0, x) because of the time-dependence of z n,κ . In other words, a small current is created from the motion of the atoms even if the macroscopic quantities remain constant in time. On Fig. 3, we displayed the deformations of the 5 lowest modulations with respect to the changes in the same potential Ṽper as in Fig. 2. According to [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a onedimensional crystalline lattice -I. Homogeneous problems[END_REF], §3.2, this implies the dragging effect suggested in [START_REF] Sundaram | Wave-packet dynamics in slowly perturbed crystals: gradient corrections and Berry-phase effects[END_REF].

E n (t, κ) = Ê0 (t) 2 + q∈N * Êq (t) cos(2πqκ/k), Êq (t) = 4 k 2 0 E n (t, κ) cos(2πqκ/k).dκ.

Numerical approximation with K-branch solutions

This method has been originally introduced in [START_REF] Gosse | Using K-branch entropy solutions for multivalued geometric optics computations[END_REF][START_REF] Runborg | Some new results in multiphase geometrical optics[END_REF] following the seminal work by Brenier and Corrias [START_REF] Brenier | A kinetic formulation for multibranch entropy solutions of scalar conservation laws[END_REF]. Is is based on the remark that geometrical solutions to ( 7)-( 9) can be recovered from the solving of a rather simple system of nonlinear conservation laws through the moment closure of a kinetic equation. This opens the way of using all the results from the numerical analysis of this kind of problems instead of using the classical Lagrangian framework of ray-tracing algorithms, which may ask for repetitive regridding procedures. Let us also note that the inclusion of the Poisson self-consistent potential wouldn't be easy within a ray-tracing approach. We don't plan to recall it completely instead we refer to [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a onedimensional crystalline lattice -I. Homogeneous problems[END_REF][START_REF] Gosse | Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice -II. Impurities, confinement and Bloch oscillations[END_REF] where it has been explained how to extend it to the case where semi-classical dynamics are driven by energy bands obtained from Bloch's decomposition. The inclusion of the Poisson term for systems endowed with translational invariance in two directions has been described in [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a onedimensional crystalline lattice -III. From ab-initio models to WKB for Schrödinger-Poisson[END_REF] and will be used in §4.3. Here we just explain how to handle the time-dependence of the energy bands in the processing of the WKB system (7)-(9).

General procedure

It has been observed [START_REF] Brenier | Averaged multivalued solutions for scalar conservation laws[END_REF] that the geometric solutions of scalar conservation laws with a flux E n (., u) ≥ 0 can be recovered out of a kinetic problem,

∂ t f + E n (t, ξ)∂ x f = 0, f (t = 0, x, ξ) = H(u(t = 0, x) -ξ)H(ξ),
with H the Heaviside function. Beyond breakup time, the form of f must express the fact that several particles with different velocities can cross at the same point x. Thus a more correct representation would be,

f (t, x, ξ) = K j=1 (-1) j-1 H(u j (t, x) -ξ), (11)
as long as no more than K folds appear. A remarkable feature is that (11) can be obtained from an entropy minimization process; this eventually led to the definition of K-multivalued solutions in [START_REF] Brenier | A kinetic formulation for multibranch entropy solutions of scalar conservation laws[END_REF]:

Definition 1 We call K-multivalued solution any measurable function f (t, x, ξ) ∈ {0, 1} on R × R + × R + satisfying the following equation in the sense of distributions

∂ t f + E n (t, ξ)∂ x f = (-1) K-1 ∂ K ξ µ, f (t, x, ξ) = K j=1 (-1) j-1 H(u j (t, x) -ξ), ( 12
)
where µ is a nonnegative Radon measure on R × R + × R + and K ∈ N is given.

The set of u j (t, x)'s is called the K-branch entropy solution; as usual, moments

m i (t, x) = 1 i K j=1 (-1) j-1 u j (t, x) i-1 , i = 1, 2, .
.., K can be computed, for which an equivalence result holds:

Theorem 1 (Brenier & Corrias, [6]) A measurable function f (t, x, ξ) = K j=1 (-1) j-1 H(u j (t, x) -ξ) is a K-multivalued solution if
and only if all the following entropy inequalities hold for any θ, ∂ K ξ θ ≥ 0:

∂ t R + θ(ξ)f (t, x, ξ).dξ + ∂ x R + E n (t, ξ)θ(ξ)f (t, x, ξ).dξ ≤ 0. ( 13
)
Equality holds in case ∂ K ξ θ ≡ 0, especially for θ(ξ) = ξ j , j = 0, 1, ..., K -1. We close this section mentioning that the map R K m → u is called the "finite Markov moment problem"; this delicate inversion has been recently tackled in [START_REF] Gosse | Finite moment problems and applications to multiphase computations in geometric optics[END_REF].

Moment systems and intensity recovery

Using fft routines in 2-D (t and κ or y), it is possible to obtain numerically good approximations of the following periodic functions, (T per < +∞)

[0, T per ] × B t, κ → E n (t, κ), and, [0, T per ] × 0, 2π k × B t, y, κ → z n,κ (t, y).
The fact that we restricted ourselves to commensurate crystals is fundamental here. These two representations wouldn't hold in Fourier space without assuming that Ṽper is periodic in both its variables. This leads to restrictions on the k's allowed to show up in [START_REF] Bao | On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime[END_REF]. If one introduces a velocity variable u = ∂ x ϕ in [START_REF] Carles | Semiclassical asymptotics for weakly nonlinear Bloch waves[END_REF], then a scalar conservation law with a time-dependent flux appears,

∂ t u + ∂ x E n (t, u) = 0, u(t = 0, .) = ∂ x ϕ(t = 0, .),
for which the multivalued (or geometric) solution is to be sought according to the moment system [START_REF] Gosse | Finite moment problems and applications to multiphase computations in geometric optics[END_REF]. As soon as one completes this program, the principal intensity |a 0 | 2 can be easily recovered; indeed, at any time t > 0, one deduces from (9) that

|a 0 | 2 (t, x) = |a 0 | 2 (t = 0, x 0 ) ∂x 0 ∂x ,
and from ( 7) that x = x 0 + t 0 E n (s, u(t = 0, x 0 )).ds = x 0 + t 0 E n (s, u(t, x)).ds with u(t, x) supposedly known from the solving of (13). In the homogeneous case, the most accurate way to derive the intensity follows from

∂x 0 ∂x = ∂x ∂x 0 -1 = 1 1 + ∂ x u(t = 0, x 0 ) t 0 E n (s, u(t, x)).ds
, which leads to the expression:

|a 0 | 2 (t, x) = a 0 (t = 0, x 0 ) 1 + ∂ x u(t = 0, x 0 ) t 0 E n (s, u(t, x)).ds 2 , x 0 = x- t 0 E n (s, u(t, x)).ds.

Numerical results

We now aim at showing results of K-branch solutions in the context of simple phonon scattering via the WKB approach ( 7)-( 9) and to compare them to direct computations of solutions to (4) using the time-splitting Fourier schemes proposed in [START_REF] Bao | On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime[END_REF]. We shall try to show that the smaller ε, the closer the solutions' observables become. However, only weak convergence of observables can generally be hoped for [START_REF] Markowich | A Wigner-function approach to semiclassical limits: electrons in a periodic potential[END_REF], hence we shall look at the L 1 (R) norm of the antiderivative of the difference between position densities:

x → x 0 ε W KB (T, s) -|ψ(T, s)| 2 .ds, T ∈ R + . (14)
This function can be expected to flatten as ε is decreased. ε W KB stands for the position density obtained from the WKB ansatz (10). It's been sometimes necessary to filter numerically the Fourier schemes; here we used a standard convolution receipe involving a Gaussian kernel exp(-aξ 2 ), a ∈ R + . 512 discretization points have been used for both algorithms.

One wavenumber

We first consider the perturbed potential which is 4π-periodic:

Ṽper (t, y) = cos y - π 6 sin(ky -t) , ω = k = 1 2 . ( 15
)
The corresponding Brillouin zone is therefore B = [-1 4 , 1 4 ] and Fig. 4 displays both the first conduction band and its associated modulation |z n,κ=0 | 2 . The n index corresponds to the first conduction band, namely the first one with positive effective mass lying partially above the potential well. As time t goes by, the band is slightly deformed and not just translated. The initial data is of the form [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a onedimensional crystalline lattice -I. Homogeneous problems[END_REF] with ϕ(t = 0, x) = cos(x -π)/10 and |a 0 | 2 (t = 0, x) = exp(-(x -π) 2 )/π. We iterated up to T = 5 and the comparison between position densities is shown in Fig. 5 for ε = 1/35. The agreement is very satisfying and one even observes the weak convergence as ε decreases from 1/5 to 1/35 by checking the L 1 norm of ( 14). The following perturbed potential is still 4π-periodic, but slightly more involved, Ṽper (t, y) = cos x -π 6 0.7 sin(ky -t) + 0.3 cos(ky -t) ,

where ω = k = 1 2 . Results in T = 5 are shown in Fig. 6; once again, the agreement between both computations can be considered satisfying especially for ε = 1/47.

Two different wavenumbers

We switch to the perturbed potential which is 12π-periodic:

Ṽper (t, y) = cos x -π 10 sin(y/2 -t) + sin(y/6 -Ω(1/6)t) .

This case is difficult because the Brillouin zone is very small (k 1 = 1 6 and k 2 = 1 2 ) thus (by smoothness) the bands are nearly flat and one must iterate for a long time to observe the dynamics. Fig. 7 displays the outcome in T = 30 of both WKB approach and direct Schrödinger computations from the initial data [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a onedimensional crystalline lattice -I. Homogeneous problems[END_REF] with ϕ(t = 0, x) = cos(x -π)/12 and |a 0 | 2 (t = 0, x) = exp(-(x -π) 2 )/π. We chosed the two vibration modes k = 1 2 and k = 1 6 because the resulting Ṽper is still a periodic function in y and t. Periodicity in t is important if one doesn't want to store all values of E n involved in the computation. The agreement between position densities is less clear in this case, however the two central spikes are correctly located. 3-branch solutions are very good approximations of the ray-tracing picture, see again Fig. 7.

Inclusion of the Poisson potential

We move back to the simple perturbed potential (15) but we include now repulsive self-interaction effects between electrons. That is to say, we add a term V P (t, x)ψ on the right-hand side of (4), where -∂ xx V P = |ψ| 2 . As a consequence, it hasn't been possible to obtain a breakup and multivalued K-branch solutions for K > 1. The perturbed potential and the initial data are the same than in §4.1 but iterations went only up to T = 3. It wasn't possible to display a ray-tracing solution in this weakly nonlinear case, thus we just observe a quite good adequation between position densities on Fig. 8. The L 1 (R) norm of ( 14) decreases slowly indeed with ε → 0, as shown in Fig. 9, whereas strong L 1 convergence doesn't seem to be visible.

Conclusion

We have proposed an original two-scale WKB technique adapted to electronic conduction in 1-D harmonic crystals, see ( 7)-( 9)- [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a onedimensional crystalline lattice -I. Homogeneous problems[END_REF]. A numerical processing has been presented further together with a validation against direct Schrödinger computations. The main shortcoming in this general approach is that in case B becomes really tiny (that is to say, many small wave numbers are present in (2)), band gaps are likely to shrink so much as to allow electrons to perform interband transitions rather easily, thus a mixed state

I i=1 λ i ψ ε ni (t, x), λ i ≥ 0, I i=1 λ i = 1,
with I = 1/k for instance, should be better considered (like in [START_REF] Bechouche | Semiclassical limit for the Schrödinger-Poisson equation in a crystal[END_REF] for steady potentials) instead of the pure state [START_REF] Gosse | Multiphase semiclassical approximation of an electron in a onedimensional crystalline lattice -I. Homogeneous problems[END_REF]. Still, our ansatz ( 10) is more precise compared to standard phonon modelling as a collision term in a kinetic equation. Indeed, one central hypothesis in this last approach is to assume that the resulting scattering effects are instantaneous; in other terms, ε = 0 and an adiabatic decoupling takes place. A major open problem lies in extending our analysis in §2 to non-periodic potentials, this would match the so-called incommensurate quasicrystals, [START_REF] De Lange | Electrons in incommensurate crystals: spectrum and localization[END_REF][START_REF] Quilichini | Phonon excitations in quasicrystals[END_REF].

Figure 1 .

 1 Figure 1. Time evolution for periodic potential in (4) with Vper(y) = cos(y) and u(t, y) = sin(ky + Ω(k)t), k 0.02, ε = 1.

Figure 2 .

 2 Figure 2. Two lowest modulations |z n,κ=0 | 2 for k = 0.5 and t = 0.1.

Figure 3 .

 3 Figure 3. Potentials (left) and lowest modulations |z n,κ=0 | 2 (right) for t = 0, 0.3.

Figure 4 .

 4 Figure 4. Time evolution of En (top) and |z n,κ=0 | 2 (bottom) for (15).

Figure 5 .

 5 Figure 5. Position density for (15) in T = 5 and ε = 1/35 (left) and weak convergence as ε → 0 (L 1 (R) norm of (14)).

Figure 6 .

 6 Figure 6. Position density for (16) in T = 5 and ε = 1/35 (top) and ε = 1/47 (bottom). Figures on the right are in log-scale.

Figure 7 . 3 -

 73 Figure 7. 3-branch velocity and intensity for (17) in T = 30 (top) and corresponding position density for ε = 1/26, ε = 1/50 (bottom).

Figure 8 . 3 -

 83 Figure 8. 3-branch velocity and intensity for (17) in T = 3 (top) and corresponding position density for ε = 1/35 (bottom).

Figure 9 .

 9 Figure 9. Decay of the L 1 (R) norm of (14) for ε ∈ [1/65, 1/5].

‡ Partially supported by Fundacao para Ciencia e Tecnologia, POCTI-ISFL-1-209 (Centro de Matematica e Applicacoes Fundamentais). Thanks are due to Dr. R. Carles for insightful comments.