On the singular spectrum for adiabatic quasi-periodic Schr\"odinger Operators

In this paper we study spectral properties of a family of quasi-periodic Schr\"odinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curve has a real branch that is extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent and show that the spectrum is purely singular. This result was conjectured and proved in a particular case by Fedotov and Klopp in \cite{FEKL1}.


Introduction
We consider the following Schrödinger equation where x → V (x) and ζ → W (ζ) are periodic, and ε is chosen so that the potential V (· − z) + W (ε·) be quasi-periodic. Note that in this case, the family of equations (1.1) is ergodic, see [18]; in this case so its spectrum does not depend on z, see [1]. The operator H ϕ,ε can be regarded as an adiabatic perturbation of the periodic operator H 0 : Equation (1.1) is one of the main models of solid state physics. The function ψ is the wave function of an electron in a crystal with impurities. V represents the potential of the perfect crystal; as such it is periodic. The potential W is the perturbation created by impurities. In the semiconductors, this perturbation is slow-varying with respect to the field of the crystal, [21]. It is natural to consider the semi-classical limit.
Notice that the iso-energy curves Γ C (E) and Γ R (E) are 2π-periodic in ζ and κ and Γ C is the Riemann surface uniformizing κ.
The real iso-energy curve has a well known role for adiabatic problems [2]. The adiabatic limit can be regarded as a semi-classical limit and the Hamiltonian E(κ) + W (ζ) can be interpreted as a "classical" Hamiltonian corresponding to equation (1.1).
When W has in a period exactly one maximum and one minimum, that are non-degenerate, it is proved in [9] that in the energy intervals where the adiabatic iso-energetic curves are extended along the momentum direction, the spectrum is purely singular. This result leads to the following conjecture: in a given interval, if the iso-energy curve has a real branch (a connected component) that is an unbounded vertical curve, then in the adiabatic limit, in this interval, the spectrum is singular. This paper is devoted to prove this conjecture.
Heuristically when the real iso-energy curve is extended along the momentum axis, the quantum states should be extended in momentum and, thus localized in the position space.
1.1. Results and discussions. Now, we state our assumptions and results. (H2): V is real-valued and locally square-integrable.
We define: 1.1.2. Assumptions on the energy region. To describe the energy regions where we study the spectral properties, we consider the periodic Schrödinger operator H 0 acting on L 2 (R) and defined by (1.2).
1.1.3. The periodic operator. The spectrum of (1.2) is absolutely continuous and consists of intervals of the real axis, say [E 2n+1 , E 2n+2 ] for n ∈ N , such that E 1 < E 2 ≤ E 3 < E 4 ...E 2n ≤ E 2n+1 < E 2n+2 ... and E n → +∞, n → +∞. The points (E j ) j∈N are the eigenvalues of the self-adjoint operator obtained by considering H 0 defined by (1.2) and acting in L 2 ([0, 1]) with periodic boundary conditions (see [6,19]). The intervals [E 2n+1 , E 2n+2 ], n ∈ N, are the spectral bands, and the intervals (E 2n , E 2n+1 ), n ∈ N * , the spectral gaps. When E 2n < E 2n+1 , one says that the n-th gap is open; when [E 2n−1 , E 2n ] is separated from the rest of the spectrum by open gaps, the n-th band is said to be isolated. The spectral bands and gaps are represented in figure 1.  1.2. Geometric description.
It consists of the real line and of complex branches (curves) which are symmetric with respect to the real line. There are complex branches beginning at the real extrema of W that do not cross again the real line.
Consider an extremum of W of order n i on the real line, say ζ i . Near ζ i , the set W −1 (R) consists of a real segment, and of n i − 1 complex curves symmetric with respect to the real axis and intersecting the real axis only on ζ i . The angle between two neighboring curves is equal to π n i .
We assume that Y is so small that • S Y is contained in the domain of analyticity of W ; • the set W −1 (R) ∩ S Y consists of the real line and of the complex lines passing through the real extrema of W .
An example of subset W −1 (R) is shown in figure 2.

Notations and description of
with the following properties: ] is associated to a connected component of Γ R (E) which is an unbounded vertical curve.
• We generally define: ). Let (ζ j i ) 1≤i≤p j be the extrema of W in G j . We recall that n j i is the order of ζ j i . We have the following description: There exists a finite number p of real extrema of W in [A, B].
• If p = 0, there exists Y > 0 such that: ..p} of disjoint strictly vertical lines of C + starting at ζ i such that: Here σ ac (H z,ε ) is the absolutely continuous spectrum of the family of operators (H z,ε ).

Periodic Schrödinger operators
This section is devoted to the study of the periodic Schrödinger operator (1.2) where V is a 1-periodic, real-valued, L 2 loc -function. We recall known facts needed on the present paper and we introduce notations. Basic references are [6,12,15,20].
2.1. Bloch solutions. Let ψ be a solution of the equation satisfying the relation for all x ∈ R and some non-vanishing complex number λ(E) independent of x. Such a solution exists and is called the Bloch solution and λ(E) is called Floquet multiplier. We discuss its analytic properties as a function of E.
As in section 1.1.2, we denote the spectral bands of the periodic Schrödinger operator by [E 2n+1 , E 2n+2 ], n ∈ N. Consider S ± two copies of the complex plane E ∈ C cut along the spectral bands. Paste them together to get a Riemann surface with square root branch points. We denote this Riemann surface by S. For E ∈ S, we denote by E the point on S different from E and having the same projection on C as E.
We let The function ψ(x, E) is another Bloch solution of (2.1). Except at the edges of the spectrum, the functions ψ and ψ are linearly independent solutions of (2.1). In the spectral gaps, ψ and ψ are real valued functions of x, and, on the spectral bands, they differ only by complex conjugation. All the branch point of k p are of square root type. Let E l be a branch point of k p . In a sufficiently small neighborhood of E l , the function k p is analytic in √ E − E l , and Finally, we note that the main branch can be continued analytically to the complex plane cut along (−∞, E 1 ] and the spectral gaps ]E 2n , E 2n+1 [, n ∈ N * , of the periodic operator H 0 .

2.3.
A meromorphic function. Now let us discuss a function playing an important role in the adiabatic constructions.
In [12], it is shown that, on S, there is a meromorphic function ω having the following properties: • the differential Ω = ωdE is meromorphic; its poles are the points of P ∪ Q, where P is the set of poles of E → ψ(x, E), and Q is the set of zeros of k ′ ; • all the poles of Ω are simple; • if the residue of Ω at a point p is denoted by res p Ω, one has res p Ω = 1, ∀p ∈ P \ Q, (2.5) res q Ω = −1/2, ∀q ∈ Q \ P, (2.6) • if E ∈ S projects into a gap, then ω(E) ∈ R. 6 • if E ∈ S projects inside a band, then ω(E) = ω( E).

The complex momentum.
It is the main analytic object of the complex WKB method. Let ζ ∈ S Y . We define κ, in D(W ) the domain of analyticity of W by by : Here, k is the Bloch quasi-momentum defined in section 2.2. Though κ depends on E, we omit the is a multi-valued analytic function, and its branch points are related to the branch points of the quasi-momentum by the relations Let ζ 0 be a branch point of κ. If W ′ (ζ 0 ) = 0, then ζ 0 is a branch point of square root type.
If D ⊂ D(W ) is a simply connected set containing no branch points of κ, we call it regular. Let κ p be a branch of the complex momentum analytic in a regular domain D. All the other branches that are analytic in D are described by the following formulae: Here ± and m ∈ Z are indexing the branches.

Index of an interval
We define: ] associated to κ j . Let us give some properties of p j . Lemma 2.1. Assume that (H4) is satisfied. The indexes p j have the following properties: , the index p j satisfies p j = 0. Else, we consider k j the branch of the quasi-momentum associated to κ j . We have see (2.2), and |p j | = 1.
Let us prove point (2). We write: ) are the ends of a same gap and By periodicity, (E − W )(ϕ − 1 (E)) and (E − W )(ϕ + N (E)) are the ends of the same gap. This ends the proof of Lemma 2.1.
If p j = 0, we say that we cross a band. In this case, and the associated connected component of the iso-energy curve is unbounded vertically.
Remark 2.2. We can shose the determination of κ such that p 1 = 1.

Tunneling coefficients.
For j ∈ {1, . . . , N }, we denote by γ j a smooth closed curve that goes Notice that this curve is the projection of a closed curve on the complex Riemann surface κ(ζ) = k(E − W (ζ)). We consider the tunneling actions S j given by: It is straightforward to prove that for E ∈ J, each of these actions is real and non-zero and that S j is analytic in a complex neighborhood of J (for analogous statements, we refer to [8,17]). By definition, we choose the direction of the integration so that all the tunneling actions be positive. We set T is exponentially small.
3. The Proof of Theorem 1. for sufficiently small irrational ε/2π, the Lyapunov exponent Θ(E, ε) of (1.1) is positive and satisfies the asymptotics This theorem implies that if ε/2π is sufficiently small and irrational, then, the Lyapunov exponent is positive for all E ∈ J. 8 3.1. The monodromy matrix and the Lyapunov exponents. The main object of our study in this subsection is the monodromy matrix for the family of equations (1.1), we define it briefly (we refer the reader to [8,9]). In this paper, we compute the asymptotics of its Fourier expansion in the adiabatic limit.
3.1.1. Definition of the monodromy matrix. Fix E ∈ R. Consider the family of differential equations indexed by z ∈ R, Definition 3.2. We say that (ψ i ) i∈{1,2} is a consistent basis of solutions to (3.2) if the two functions are a basis of solutions to (3.2) whose Wronskian is independent of z and that are 1-periodic in z i.e that satisfy We refer the reader to [9,12] about the existence and details on consistent basis of solutions to being also solutions of equation (3.2), we get the relation The matrix M is called the monodromy matrix associated to the consistent basis (ψ 1,2 ). We recall the following properties of this matrix.
The Matrix M belongs to SL(2, R) which is known to be isomorph to SU (1, 1).

3.2.
The Lyapunov exponents and the monodromy equation. Consider now a 1-periodic, SL(2, C)-valued function, say, z → M , and h > 0 irrational. Consider the finite difference equation Going from equation ( The Lyapunov exponent of the finite difference equation (3.6) is where the matrix cocycle (P N (z, h)) N ∈N is defined as It is well known that, if h is irrational, and M is sufficiently regular in z, then the limit (3.7) exists for almost all z and is independent of z.
We describe only one of them. Recall that Θ(E, ε) is the Lyapunov exponent of the equation (1.1) We have the following result.  The following result gives the asymptotics of A and B in the adiabatic case.  (3.11). When ε tends to 0, the coefficients A and B admit the asymptotics The integers P − , P + and Q − , Q + ∈ Z are specified in section 3. There exists a constant C > 1 such that for ε > 0 sufficiently small and E ∈ V 0 ∩ R, one has (3.14) where T (E, ε) is defined in (2.13).
that the asymptotics (3.12) and (3.13)  This leads to the following upper bound for the Lyapunov exponent for the matrix cocycle generated by M (z, E).
where C is a constant independent of E and ε. Using (3.10) one gets

The lower bound.
For (M (z, ε) 0<ε<1 ) a family of SL(2, C)-valued, 1-periodic functions of z ∈ C and h an irrational number, we recall the following result obtained in [10].
First for σ = 0 1 1 0 we prove that the matrix σM (z, E)σ completes the assumption of Proposition 3.6.
Let y 0 and y 1 fixed such that 0 < y 0 < y 1 < Y . The asymptotics of the monodromy matrix are uniform for z in S = {z ∈ C; y 0 /ε ≤ Imz ≤ y 1 /ε} and E ∈ V 0 .
Let us assume that n = N N in (5.12) is even then the following relations holds Where c(E) is independent of z and bounded by a constant uniformly in ε and E. So, we have This gives that the matrix-valued function satisfies the assumptions of Proposition 3.6.

Where c(E) is independent of z and bounded by a constant uniformly in ε and E. So, we have
The moste important properties of the matrix M (z, E) in Proposition 3.6, is that the bigger eigenvalue is 1 [10].
Using (3.20), we get that the Lyapunov exponent θ(E, ε) of the matrix cocycle associated to (M (·, E), h), satisfies the estimates Taking into account (1.2) we get that By (3.14) we get

The expression of T (E) given by (3.25) and (3.24) give (3.1) for any
Recall that V 0 ∩ R is an open interval containing E 0 ∈ J. The above construction can be carried out for any E 0 ∈ J. The end of the proof of Theorem 1.2 follows from the compactness of the interval J.

The complex WKB method for adiabatic problems
In this section, following [12,11,14], we describe the complex WKB method for adiabatically perturbed periodic Schrödinger equations Here, V is 1-periodic and real valued, ε is a small positive parameter, and the energy E is complex; one assumes that V is L 2 loc and that W is analytic in a strip in the neighborhood S Y of the real line. The parameter ζ is an auxiliary complex parameter used to decouple the slow variable ζ = εx and the fast variable x. The idea of this method is to study solutions of (4.1) in some domains of the complex plane of ζ and, then to recover information on their behavior in x ∈ R. Therefore, for D a complex domain, one studies solutions satisfying the condition: The aim of the WKB method is to construct solutions to (4.1) satisfying (4.2) and that have simple asymptotic behavior when ε tends to 0. This is possible in certain special domains of the complex plane of ζ. These domains will depend continuously on V, W and E. We shall use these solutions to 13 compute the monodromy matrix; we consider V and W as fixed and construct the WKB objects and solutions in an uniform way for energies in a neighborhood of E.
that are moreover analytic in ζ on a given regular domain.
Let ζ 0 be a regular point (i.e. ζ 0 is not a branch point of κ).
Let U 0 be a sufficiently small neighborhood of E 0 , and let V 0 be a neighborhood of ζ 0 such that In U 0 , we fix a branch of the function k ′ (E) and consider ψ ± (x, E), the two branches of the Bloch solution ψ(x, E) and Ω ± , the corresponding branches of Ω (see section 2.3. For ζ ∈ V 0 , we set The functions Ψ ± are called the canonical Bloch solutions normalized at the point ζ 0 .
The properties of the differential Ω imply that the solutions Ψ ± can be analytically continued from V 0 to any regular domain containing V 0 .
We set We call ζ 0 the normalization point for f . To say that a consistent solution f has standard behavior, we will use the following notation

4.3.
Some results on the continuation of asymptotics.

Description of the Stokes lines near
. This section is devoted to the description of the Stokes lines under assumption (H4).

4.3.2.
Definition. The definition of the Stokes lines is fairly standard, [13,8]. The integral ζ → ζ κ(u)du has the same branch points as the complex momentum. Let ζ 0 be one of them. Consider the curves beginning at ζ 0 , and described by the equation Assume that W ′ (ζ 0 ) = 0. Equation (2.4) implies that there are exactly three Stokes lines beginning at ζ 0 . The angle between any two of them at this point is equal to 2π 3 . Indeed for ζ near ζ 0 , we have (1)).

4.3.3.
Stokes lines for E 0 ∈ J. We describe the Stokes lines beginning at ϕ − 1 (E) and ϕ + 1 (E). Since W is real on R, the set of the Stokes lines is symmetric with respect to the real line.
The proof of this Lemma is similar to the studies done in [9,14,16,17]. We do not give the details.

Construction of a consistent basis near
We recall this result, proved in [14].
We mimic the analysis done in section 5 of [9]. Precisely, we start by a local construction of the solution f using canonical domain; then, we apply continuation tools i.e the rectangle Lemma, the adjacent domain principle and the Stokes Lemma.

The Proof of Theorem 3.4
The Proof of Theorem 3.4 follows the same ideas as the computations given in section 10.2 of [9]. Below we only give the details for the proof of (3.12) and (3.13).
5.1. Strategy of the computation. We now begin with the construction of the consistent basis the monodromy matrix of which we compute. Recall that (H1) − (H4) are satisfied.
In the present section, we construct and study a solution f of (3.2) satisfying 3.3.
To use the complex WKB method, we perform the following change of variable in (3.2) Then (3.2) takes the form (4.1). In the new variables, the consistency condition (3.3) becomes (4.2).
Note also that in the new variables, for two solutions to (4.1) to form a consistent basis, in addition to being a basis of consistent solutions, their Wronskian has to be independent of ζ.
The aim of this section is the computation of M (ζ, E, ε). The definition of the monodromy matrix implies that .
This gives that the monodromy matrix is analytic in ζ in the strip S Y and in E in a constant neighborhood of E 0 . By the definition of f 1 we get that A and B are ε-periodic in ζ ∈ S Y . This is a an immediate consequence of the properties of f 1 .
Therefore, we will compute the Fourier series of A and B.The strategy of the computation is based on the ideas of [9] and we first recall some notions presented there. We refer the reader to this paper for more details.
Let h and g having a standard asymptotic behavior in regular domains D h and D g and solutions of (4.1): Here, κ h , (resp. κ g ) is an analytic branch of the complex momentum in D h (resp.D g ), Ψ h (resp. Ψ g ) is the canonical Bloch solution defined on D h (resp. D g ), and ζ h (resp. ζ g ) is the normalization point for h (resp. g).
As the solutions h and g satisfy the consistency condition (4.2), their Wronskian is ε-periodic in ζ.

5.1.1.
Arcs. We assume that D g ∩ D h contains a simply connected domainD. Let γ be a regular curve going from ζ g to ζ h in the following way: staying in D g , it goes from ζ g to some point inD, then, staying in D h , it goes to ζ h . We say that γ is an arc associated to the triple h, g andD.
AsD is simply connected, all the arcs associated to the triple, h, g andD. AsD is simply connected, all the arcs associated to one and the same triple can naturally be considered as equivalent; we denote them by γ (h, g,D).

5.1.2.
The meeting domain. LetD be as above. We callD a meeting domain, if, inD, the function Im κ h and Im κ g do not vanish and are of opposite signs.
Note that, for small values of ε, whether ζ → g(x, ζ) and ζ → h(x, ζ) increase or decrease is determined by the exponential factor e κ g dζ, the action of the arc γ = γ(h, g, D). Clearly, the action takes the same value for equivalent arcs.
Assume that E(ζ) / ∈ P ∪ Q along γ(h, f,D). Consider the function q g = k ′ (E) and the 1-form Ω g (E(ζ)) in the definition of Ψ g . Continue them analytically along γ. We set A is called the the amplitude of the arc γ. The properties of Ω imply that the amplitudes of two equivalent arcs γ(h, g,D) coincide.

5.2.
Results on the Fourier coefficients. We recall the following result from [9] Proposition 5.1. Let d = d(h, g) be a meeting domain for h and g, and m = m(h, g, d) be the corresponding index. Then where w m is the constant given by Here ψ + = ψ h and ψ − is complementary to ψ + . The asymptotic (5.6) is uniform in ζ and E when ζ stays in a fixed compact of S(d) and E in a small enough neighborhood of E 0 .

5.2.1.
The index m. Let ζ 0 be a regular point. consider a regular curve γ going from ζ 0 to ζ 0 + 2π.
Let κ be a branch of the complex momentum that is continuous on γ. We call the couple (γ, κ) a period. Let (γ 1 , κ 1 ) and (γ 2 , κ 2 ) be two periods. Assume that one can continuously deform γ 1 into γ 2 without intersecting any branching point. By this we define an analytic continuation of κ 1 to γ 2 . If the analytic continuation coincide with κ 2 , we say that the periods are equivalent.
The numbers σ = σ(γ, κ) and m = m(γ, κ) are called respectively the signature and the index of the period (γ, κ). They coincide for equivalent periods.
We use Lemma 5.2 to compute the index. To do this, we have to compute Re κ at the intersection of γ 0 + 2π and G.
The computation of the index λ − j of G j gives in this case that λ − 1 = −1 + (−1) 2 .
Similarly to the computation of B, we define the index β + j of G j and in this case, we have: and β + j = (−1) N j −1 + (−1) Pl=n j l=1 (o l −1) 2 .

.
So by the above notation we get that