# On the Singular Spectrum for Adiabatic Quasiperiodic Schrödinger Operators

Authors: Magali Marx; Hatem Najar
Journal: Advances in Mathematical Physics
 (2010)

---

## Abstract

We study spectral properties of a family of quasiperiodic Schrödinger
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 (2005).


---

## Body

## 1. Introduction

We consider the following Schrödinger equation:(1.1)(Hz,εψ)(x)=-d2dx2ψ(x)+[V(x-z)+W(εx)]ψ(x)=Eψ(x),x∈ℝ,
where x↦V(x) is 1-periodic, ζ↦W(ζ) is 2π-periodic, and ε small 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 [1]; so its spectrum does not depend on z; see [2]. The operator Hz,ε can be regarded as an adiabatic perturbation of the periodic operator H0:(1.2)H0=-d2dx2+V(x).
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 an external electric field.  V represents the potential of the perfect crystal; as such it is periodic. The potential W represents an external electric field. In the semiconductors, this perturbation is slow-varying with respect to the field of the crystal, [3]. It is natural to consider the semiclassical limit.The Iso-Energy Curve
Letℰ(κ) be the dispersion relation associated to H0. Consider the complex and real-isoenergy curves  Γℂ and Γℝ defined by
(1.3)Γℂ(E)={(κ,ζ)∈ℂ2;ℰ(κ)+W(ζ)=E};Γℝ(E)={(κ,ζ)∈ℝ2;ℰ(κ)+W(ζ)=E}.
Notice that the iso-energy curves Γℂ(E) and Γℝ(E) are 2π-periodic in ζ and κ and Γℂ is the Riemann surface uniformizing κ.The real iso-energy curve has a well-known role for adiabatic problems [4]. The adiabatic limit can be regarded as a “semiclassical” limit and the Hamiltonian ℰ(κ)+W(ζ) can be interpreted as a “classical” Hamiltonian corresponding to (1.1).In the case when the intervalE-W(ℝ) contains one spectral band (we refer to that by “isolated band” model), the iso-energy curve is presented in Figure 1. The real branches are vertical curves, they are connected by complex loops (closed curves) lying on Γℂ; loops are represented by horizontal closed curves. In this case the connected components are extended in κ-direction and bounded in ζ-direction.Figure 1
The iso-energy curve for “isolated band” model.In the case when the intervalE-W(ℝ) is contained in a spectral band (we refer to that by the “band middle” model), the iso-energy curve is presented in Figure 2. The horizontal curves are connected components of Γℝ; the vertical loops are situated in Γℂ. In this case the connected components are bounded in κ direction and extended in ζ-direction.Figure 2
The iso-energy curve for “band middle” model.WhenW has in a period exactly one maximum and one minimum, that are nondegenerate, it is proved in [5] 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 of the real iso-energy curve Γℝ, see Figure 1 and [5]) 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.


#### 1.1.1. Assumptions on the Potentials

We assume the following.(H1)
V and W are periodic:
(1.4)V(x+1)=V(x),W(x+2π)=W(x),∀x∈ℝ.(H2)
V is real-valued and locally square-integrable.(H3)
W is real analytic in the strip SY={z∈ℂ;|Imz|<Y}.We define(1.5)W-=inft∈ℝW(t),W+=supt∈ℝW(t).
We define the spectral window 𝒲(E) by (1.6)𝒲(E)=E-W(ℝ)=[E-W+,E-W-].


#### 1.1.2. Assumptions on the Energy Region

To describe the energy regions where we study the spectral properties, we consider the periodic Schrödinger operatorH0 acting on L2(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 [E2n+1,E2n+2] for n∈ℕ, such that E1<E2≤E3<E4,…,E2n≤E2n+1<E2n+2,… and En→+∞,n→+∞. The points (Ej)j∈ℕ are the eigenvalues of the self-adjoint operator obtained by considering H0 defined by (1.2) and acting in L2([0,2]) with periodic boundary conditions (see [6, 7]). The intervals [E2n+1,E2n+2],n∈ℕ, are the spectral bands, and the intervals (E2n,E2n+1),n∈ℕ*, the spectral gaps. When E2n<E2n+1, one says that the nth gap is open; when [E2n-1,E2n] is separated from the rest of the spectrum by open gaps, the nth band is said to be isolated. The spectral bands and gaps are represented in Figure 4.


#### 1.1.4. The Geometric Assumption

Let us describe the energy region where we study (1.1). We assume that J is a real compact interval such that, for all E∈J, the window 𝒲(E) contains exactly N≥1 isolated bands of the periodic operator. So, we fix two positive integers n and N>0 and assume that:  (H4a)
The bands[E2(n+j)-1,E2(n+j)],j=1,2,…,N, are isolated.(H4b)
For allE∈J, these bands are contained in the interior of 𝒲(E).(H4c):
For allE∈J, the rest of the spectrum of H0 is outside of 𝒲(E).Remark 1.1.
The geometric assumption assures that the iso-energy curveΓℝ(E) contains a real branch that is an unbounded vertical curve.
We asked that the window contains only isolated bands of the periodic operator to have a control on the branch points of the Bloch quasimomentum and on its properties of analyticities.


#### 1.1.5. The Main Result

The main object of this paper is to prove the following.Theorem 1.2.
LetJ be a real compact interval. We assume that (H1)–(H4) are satisfied. For ε>0 sufficiently small, for almost all z∈ℝ, one has
(1.7)σac(Hz,ε)∩J=∅.
Here σac(Hz,ε) is the absolutely continuous spectrum of the family of (Hz,ε).Remark 1.3.
(1)It is proved in [8] that for Σ=σ(H0)+W(ℝ)=σ(H0)+[W-,W+], one has the following for all ε≥0,σ(Hz,ε)⊂Σ, and for any K⊂Σ,  compact, there exists C>0 such that for all ε sufficiently small and all E∈K,
(1.8)σ(Hz,ε)∩(E-Cε1/2,E+Cε1/2)≠∅.
So, for an interval J as in Theorem 1.2, we have
(1.9)σ(Hz,ε)∩J≠∅.
(2) Using the Ishii-Pastur-Kotani Theorem [1, 9], one can see that the result of Theorem 1.2 is deduced from the positivity of the Lyapunov exponent. This will be done by computing the asymptotics for the Lyapunov exponent. With the aim of simplifying the introduction, we do not give it here; it is the subject of Section 3.1.


## 1.1. Results and Discussions

Now, we state our assumptions and results.


### 1.1.1. Assumptions on the Potentials

We assume the following.(H1)
V and W are periodic:
(1.4)V(x+1)=V(x),W(x+2π)=W(x),∀x∈ℝ.(H2)
V is real-valued and locally square-integrable.(H3)
W is real analytic in the strip SY={z∈ℂ;|Imz|<Y}.We define(1.5)W-=inft∈ℝW(t),W+=supt∈ℝW(t).
We define the spectral window 𝒲(E) by (1.6)𝒲(E)=E-W(ℝ)=[E-W+,E-W-].


### 1.1.2. Assumptions on the Energy Region

To describe the energy regions where we study the spectral properties, we consider the periodic Schrödinger operatorH0 acting on L2(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 [E2n+1,E2n+2] for n∈ℕ, such that E1<E2≤E3<E4,…,E2n≤E2n+1<E2n+2,… and En→+∞,n→+∞. The points (Ej)j∈ℕ are the eigenvalues of the self-adjoint operator obtained by considering H0 defined by (1.2) and acting in L2([0,2]) with periodic boundary conditions (see [6, 7]). The intervals [E2n+1,E2n+2],n∈ℕ, are the spectral bands, and the intervals (E2n,E2n+1),n∈ℕ*, the spectral gaps. When E2n<E2n+1, one says that the nth gap is open; when [E2n-1,E2n] is separated from the rest of the spectrum by open gaps, the nth band is said to be isolated. The spectral bands and gaps are represented in Figure 4.


### 1.1.4. The Geometric Assumption

Let us describe the energy region where we study (1.1). We assume that J is a real compact interval such that, for all E∈J, the window 𝒲(E) contains exactly N≥1 isolated bands of the periodic operator. So, we fix two positive integers n and N>0 and assume that:  (H4a)
The bands[E2(n+j)-1,E2(n+j)],j=1,2,…,N, are isolated.(H4b)
For allE∈J, these bands are contained in the interior of 𝒲(E).(H4c):
For allE∈J, the rest of the spectrum of H0 is outside of 𝒲(E).Remark 1.1.
The geometric assumption assures that the iso-energy curveΓℝ(E) contains a real branch that is an unbounded vertical curve.
We asked that the window contains only isolated bands of the periodic operator to have a control on the branch points of the Bloch quasimomentum and on its properties of analyticities.


### 1.1.5. The Main Result

The main object of this paper is to prove the following.Theorem 1.2.
LetJ be a real compact interval. We assume that (H1)–(H4) are satisfied. For ε>0 sufficiently small, for almost all z∈ℝ, one has
(1.7)σac(Hz,ε)∩J=∅.
Here σac(Hz,ε) is the absolutely continuous spectrum of the family of (Hz,ε).Remark 1.3.
(1)It is proved in [8] that for Σ=σ(H0)+W(ℝ)=σ(H0)+[W-,W+], one has the following for all ε≥0,σ(Hz,ε)⊂Σ, and for any K⊂Σ,  compact, there exists C>0 such that for all ε sufficiently small and all E∈K,
(1.8)σ(Hz,ε)∩(E-Cε1/2,E+Cε1/2)≠∅.
So, for an interval J as in Theorem 1.2, we have
(1.9)σ(Hz,ε)∩J≠∅.
(2) Using the Ishii-Pastur-Kotani Theorem [1, 9], one can see that the result of Theorem 1.2 is deduced from the positivity of the Lyapunov exponent. This will be done by computing the asymptotics for the Lyapunov exponent. With the aim of simplifying the introduction, we do not give it here; it is the subject of Section 3.1.


## 1.1.1. Assumptions on the Potentials

We assume the following.(H1)
V and W are periodic:
(1.4)V(x+1)=V(x),W(x+2π)=W(x),∀x∈ℝ.(H2)
V is real-valued and locally square-integrable.(H3)
W is real analytic in the strip SY={z∈ℂ;|Imz|<Y}.We define(1.5)W-=inft∈ℝW(t),W+=supt∈ℝW(t).
We define the spectral window 𝒲(E) by (1.6)𝒲(E)=E-W(ℝ)=[E-W+,E-W-].


## 1.1.2. Assumptions on the Energy Region

To describe the energy regions where we study the spectral properties, we consider the periodic Schrödinger operatorH0 acting on L2(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 [E2n+1,E2n+2] for n∈ℕ, such that E1<E2≤E3<E4,…,E2n≤E2n+1<E2n+2,… and En→+∞,n→+∞. The points (Ej)j∈ℕ are the eigenvalues of the self-adjoint operator obtained by considering H0 defined by (1.2) and acting in L2([0,2]) with periodic boundary conditions (see [6, 7]). The intervals [E2n+1,E2n+2],n∈ℕ, are the spectral bands, and the intervals (E2n,E2n+1),n∈ℕ*, the spectral gaps. When E2n<E2n+1, one says that the nth gap is open; when [E2n-1,E2n] is separated from the rest of the spectrum by open gaps, the nth band is said to be isolated. The spectral bands and gaps are represented in Figure 4.


## 1.1.4. The Geometric Assumption

Let us describe the energy region where we study (1.1). We assume that J is a real compact interval such that, for all E∈J, the window 𝒲(E) contains exactly N≥1 isolated bands of the periodic operator. So, we fix two positive integers n and N>0 and assume that:  (H4a)
The bands[E2(n+j)-1,E2(n+j)],j=1,2,…,N, are isolated.(H4b)
For allE∈J, these bands are contained in the interior of 𝒲(E).(H4c):
For allE∈J, the rest of the spectrum of H0 is outside of 𝒲(E).Remark 1.1.
The geometric assumption assures that the iso-energy curveΓℝ(E) contains a real branch that is an unbounded vertical curve.
We asked that the window contains only isolated bands of the periodic operator to have a control on the branch points of the Bloch quasimomentum and on its properties of analyticities.


## 1.1.5. The Main Result

The main object of this paper is to prove the following.Theorem 1.2.
LetJ be a real compact interval. We assume that (H1)–(H4) are satisfied. For ε>0 sufficiently small, for almost all z∈ℝ, one has
(1.7)σac(Hz,ε)∩J=∅.
Here σac(Hz,ε) is the absolutely continuous spectrum of the family of (Hz,ε).Remark 1.3.
(1)It is proved in [8] that for Σ=σ(H0)+W(ℝ)=σ(H0)+[W-,W+], one has the following for all ε≥0,σ(Hz,ε)⊂Σ, and for any K⊂Σ,  compact, there exists C>0 such that for all ε sufficiently small and all E∈K,
(1.8)σ(Hz,ε)∩(E-Cε1/2,E+Cε1/2)≠∅.
So, for an interval J as in Theorem 1.2, we have
(1.9)σ(Hz,ε)∩J≠∅.
(2) Using the Ishii-Pastur-Kotani Theorem [1, 9], one can see that the result of Theorem 1.2 is deduced from the positivity of the Lyapunov exponent. This will be done by computing the asymptotics for the Lyapunov exponent. With the aim of simplifying the introduction, we do not give it here; it is the subject of Section 3.1.


## 2. 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, Lloc2- function. We recall known facts needed on the present paper and we introduce notations. Basic references are [6, 10–12].


### 2.1. Geometric Description

#### 2.1.1. The SetW-1(ℝ)

AsE∈ℝ, the set (E-W)-1(ℝ) coincides with W-1(ℝ). It is 2π-periodic. 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 ofW of order ni on the real line, say ζi. Near ζi, the set W-1(ℝ) consists of a real segment and of ni-1 complex curves symmetric with respect to the real axis and intersecting the real axis only on ζi. The angle between two neighboring curves is equal to π/ni. Let Y>0. We set SY={-Y≤Imζ≤Y}. We assume that Y is so small that  (i)
SY is contained in the domain of analyticity of W;(ii)
the setW-1(ℝ)∩SY consists of the real line and of the complex lines passing through the real extrema of W.An example of subsetW-1(ℝ) is shown in Figure 3.Figure 3
A subset ofW-1(ℝ)∩SY.Figure 4
Bands, gaps, and the action of mapping of the quasimomentumkp on some curves.


#### 2.1.2. Notations and Description of(E-W)-1(σ(H0))

For allE∈J, we write (2.1)(E-W)-1(σ(H0))∩ℝ=⋃k∈ℤ⋃j=1N{[φj-(E),φj+(E)]+2kπ},
with the following properties.  (i)
φ1-(E)<φ1+(E)<φ2-(E)<⋯<φN-(E)<φN+(E),0<φN+(E)-φ1-(E)<2π.(ii)
As we deal with the case when the iso-energy curve has a real branch that is extended along the momentum direction, without loss of generality we consider that the connected component[φ1-(E),φ1+(E)] is associated to a connected component of Γℝ(E) which is an unbounded vertical curve. See Remark 2.3.(iii)
We generally define the following(2.2)φj+k+N-(E)=φj-(E)+2kπ,∀j∈{1,…,N},∀k∈ℤ.
We set ℬj(E) = [φj-(E),φj+(E)],𝒵 = ∪j=1Nℬj(E),𝒢j(E) = ]φj+(E),φj+1-(E)[, and 𝒢N(E) =  (φN+(E),φ1-(E)+2π) = (φN+(E),2π)∪(2π,2π+φ1-(E)). Let (ζij)1≤i≤pj be the extrema of W in 𝒢j. We recall that nij is the order of ζij.We have the following description.Lemma 2.1.
Fix[A,B] a compact interval of ℝ.
There exists a finite numberp of real extrema of W in [A,B].    (i)
Ifp=0, there exists Y>0 such that(2.3)(E0-W)-1(ℝ)∩{ζ∈SY;Reζ∈[A,B]}=[A,B].
(ii)
For p>0, one denotes by {ζ1,…,ζp} the real extrema of W in [A,B]. There exists Y>0 and a sequence {Σi1,…,Σini-1}i∈{1…p} of disjoint and strictly vertical lines of ℂ+ starting at ζi such that(2.4)(E0-W)-1(ℝ)∩{ζ∈SY;Reζ∈[A,B]}=[A,B]⋃i=1p(⋃k=1ni-1(Σik∪Σik¯)).


### 2.2. Bloch Solutions

Letψ be a solution of the equation(2.5)-d2dx2ψ(x,ℰ)+V(x)ψ(x,ℰ)=ℰψ(x,ℰ),x∈ℝ
satisfying the relation(2.6)ψ(x+1,ℰ)=λ(ℰ)ψ(x,ℰ)
for all x∈ℝ and some nonvanishing 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 Section1.1.2, we denote the spectral bands of the periodic Schrödinger operator by [E2n+1,E2n+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.One can construct a Bloch solutionψ(x,ℰ) meromorphic on 𝒮. It is normalized by the condition ψ(1,ℰ)≡1. The poles of this solution are located in the open spectral gaps or at their edges; the closure of each spectral gap contains exactly one pole that, moreover, is simple. It is located either on 𝒮+ or on 𝒮-. The position of the pole is independent of x.Forℰ∈𝒮, we denote by ℰ̂ the point on 𝒮 different from ℰ and having the same projection on ℂ as ℰ. We let (2.7)ψ̂(x,ℰ)=ψ(x,ℰ̂),ℰ̂∈𝒮.
The function ψ̂(x,ℰ) is another Bloch solution of (2.5). Except at the edges of the spectrum, the functions ψ and ψ̂ are linearly independent solutions of (2.5). In the spectral gaps, ψ and ψ̂ are real-valued functions of x, and, on the spectral bands, they differ only by complex conjugation.


### 2.3. The Bloch Quasimomentum

Consider the Bloch solutionψ(x,ℰ). The corresponding Floquet multiplier λ(ℰ) is analytic on 𝒮. Represent it in the form λ(ℰ)=exp(ik(ℰ)). The function ℰ↦k(ℰ) is the Bloch Quasimomentum of H0. Its inverse k↦E(k) is the dispersion  relation of H0. A branching point ζ is a point where k′(ζ)=0.LetD be a simply connected domain containing no branch point of the Bloch Quasimomentum. In D, one can fix an analytic single-valued branch of k, say k0. All the other single-valued branches of k that are analytic in D are related to k0 by the following formulae:
(2.8)k(ℰ)=±k0(ℰ)+2πl,l∈ℤ.
Consider ℂ+, the upper half plane of the complex plane. On ℂ+, one can fix a single-valued analytic branch of the Quasimomentum continuous up to the real line. It can be determined uniquely by the conditions Rek(ℰ+i0)=0 and Imk(ℰ+i0)>0 for ℰ<E1. We call this branch the main branch of the Bloch Quasimomentum and denote it by kp.The functionkp conformally maps ℂ+ onto the first quadrant of the complex plane cut at compact vertical slits starting at the points πl,l∈ℕ. It is monotonically increasing along the spectral zones so that [E2n-1,E2n], the nth spectral band, is mapped on the interval [π(n-1),πn]. Along any open gap, Rekp(ℰ+i0) is constant, and Imkp(ℰ+i0) is positive and has only one nondegenerate maximum.Considerℂ0, the complex plane cut along the spectral the real line from E1 to +∞. In Figure 4, we drew two curves in ℂ0 and their images under the transformation ℰ↦kp(ℰ).All the branch points ofkp are of square root type. Let El be a branch point of kp. In a sufficiently small neighborhood of El, the function kp is analytic in E-El, and(2.9)kp(ℰ)-kp(El)=clℰ-El+o(ℰ-El),cl≠0.
Finally, we note that the main branch can be continued analytically to the complex plane cut along (-∞,E1] and the spectral gaps ]E2n,E2n+1[,n∈ℕ*, of the periodic operator H0.


### 2.4. A Meromorphic Function

Now let us discuss a function playing an important role in the adiabatic constructions.In [10], it is shown that, on 𝒮, there is a meromorphic function ω having the following properties:  (i)
the differentialΩ=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′;(ii)
all the poles ofΩ are simple;(iii)
if the residue ofΩ at a point p is denoted by respΩ, one has
(2.10)respΩ=1,∀p∈P∖Q,resqΩ=-12,∀q∈Q∖P,resrΩ=12,∀r∈P∩Q.(iv)
ifℰ∈𝒮 projects into a gap, then ω(ℰ)∈ℝ.(v)
ifℰ∈𝒮 projects inside a band, then ω(ℰ)¯=ω(ℰ̂).


#### 2.4.1. The Complex Momentum

It is the main analytic object of the complex WKB method. Letζ∈SY. We define κ, in 𝒟(W) the domain of analyticity of W, by(2.11)κ(ζ)=k(E-W(ζ)).
Here, k is the Bloch Quasimomentum defined in Section 2.3. Though κ depends on E, we omit the E-dependence. Relation (2.11) translates the properties of k into properties of κ. Hence, ζ↦κ(ζ) is a multivalued analytic function, and its branch points are related to the branch points of the Quasimomentum by the relations(2.12)E-W(ζ)=El,l=1,2,3,….
Let ζ0 be a branch point of κ. If W'(ζ0)≠0, then ζ0 is a branch point of square root type.IfD⊂𝒟(W) is a simply connected set containing no branch points of κ, we call it regular. Let κp 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:(2.13)κm±=±κp+2πm.
Here ± and m∈ℤ are indexing the branches.


#### 2.4.2. Index of an Interval[φj-(E),φj+(E)]

Fixj∈{1,…,N}. Fix a continuous branch κj of the complex momentum on [φj-(E),φj+(E)]. We define(2.14)κj+=κj(φj+);κj-=κj(φj-);pjπ=κj+-κj-.pj is the index of [φj-(E),φj+(E)] associated to κj.Let us give some properties ofpj.Lemma 2.2.
Assume that(H4) is satisfied. The indices pj have the following properties.  (1)
Forj∈{1,…,N},pj∈{-1,0,1}.(2)
∑j=1N|pj|∈2ℕ.Proof.
The points(E-W)(φj-(E)) and (E-W)(φj+(E)) are the ends of a band of σ(H0): they are distinct or coincide. If they coincide, that is, if (E-W)(φj-(E))=(E-W)(φj+(E)), the index pj satisfies pj=0. Else, we consider kj the branch of the Quasimomentum associated to κj. We have that
(2.15)|kj((E-W)(φj-(E)))-kj((E-W)(φj+(E)))|=π,
see (2.8), and |pj|=1.
Let us prove point(2).As for any j∈{1,…,N}, we have |pj|=pjmod2.  We write
(2.16)∑j=1N|pj|≡∑j=1Npjmod2,∑j=1Npj=∑j=1Nκj(φj+)-κj(φj-)π=κN(φN+)-κ1(φ1-)π+∑j=1N-1κj+1(φj+1-)-κj(φj+)π.
For j∈{1,…,N-1},(E-W)(φj+1-(E)) and (E-W)(φj+(E)) are the ends of a same gap and
(2.17)kj+1((E-W)(φj+1-(E)))≡kj((E-W)(φj+(E)))[2π].
By periodicity, (E-W)(φ1-(E)) and (E-W)(φN+(E)) are the ends of the same gap.
This ends the proof of Lemma2.2.Ifpj≠0, we say that we cross a band. In this case, (E-W)(φj-)≠(E-W)(φj+) and the associated connected component of the iso-energy curve is unbounded vertically.We notice thatp1≠0, and thus ∑j=1N|pj|>0.Remark 2.3.
We can choose the determination ofκ such that p1=1.


#### 2.4.3. Tunneling Coefficients

Forj∈{1,…,N}, we denote by γj a smooth closed curve that goes once around [φj+(E),φj+1-(E)]. 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 Sj given by(2.18)Sj(E)=i∮γjκ(ζ)dζ,∀j∈{1,…N}.
It is straightforward to prove that for E∈J, each of these actions is real and nonzero and that Sj(E) is analytic in a complex neighborhood of J (for analogous statements, we refer to [13, 14]). By definition, we choose the direction of the integration so that all the tunneling actions is positive. We set (2.19)tj=e-(1/2ε)Sj(E).Sj(E) is called the tunneling action; we choose the branch κ so that on J,Sj(E) is positive. We define (2.20)T(E,ε)=∏j=1Ntj.
So we get
(2.21)T(E,ε)=e-∑j=1NSj(E)/2ε.Sj(E) being positive we get that T is exponentially small. For more details on the properties of tunneling coefficients, see [5, Section  10].


## 2.1. Geometric Description

### 2.1.1. The SetW-1(ℝ)

AsE∈ℝ, the set (E-W)-1(ℝ) coincides with W-1(ℝ). It is 2π-periodic. 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 ofW of order ni on the real line, say ζi. Near ζi, the set W-1(ℝ) consists of a real segment and of ni-1 complex curves symmetric with respect to the real axis and intersecting the real axis only on ζi. The angle between two neighboring curves is equal to π/ni. Let Y>0. We set SY={-Y≤Imζ≤Y}. We assume that Y is so small that  (i)
SY is contained in the domain of analyticity of W;(ii)
the setW-1(ℝ)∩SY consists of the real line and of the complex lines passing through the real extrema of W.An example of subsetW-1(ℝ) is shown in Figure 3.Figure 3
A subset ofW-1(ℝ)∩SY.Figure 4
Bands, gaps, and the action of mapping of the quasimomentumkp on some curves.


### 2.1.2. Notations and Description of(E-W)-1(σ(H0))

For allE∈J, we write (2.1)(E-W)-1(σ(H0))∩ℝ=⋃k∈ℤ⋃j=1N{[φj-(E),φj+(E)]+2kπ},
with the following properties.  (i)
φ1-(E)<φ1+(E)<φ2-(E)<⋯<φN-(E)<φN+(E),0<φN+(E)-φ1-(E)<2π.(ii)
As we deal with the case when the iso-energy curve has a real branch that is extended along the momentum direction, without loss of generality we consider that the connected component[φ1-(E),φ1+(E)] is associated to a connected component of Γℝ(E) which is an unbounded vertical curve. See Remark 2.3.(iii)
We generally define the following(2.2)φj+k+N-(E)=φj-(E)+2kπ,∀j∈{1,…,N},∀k∈ℤ.
We set ℬj(E) = [φj-(E),φj+(E)],𝒵 = ∪j=1Nℬj(E),𝒢j(E) = ]φj+(E),φj+1-(E)[, and 𝒢N(E) =  (φN+(E),φ1-(E)+2π) = (φN+(E),2π)∪(2π,2π+φ1-(E)). Let (ζij)1≤i≤pj be the extrema of W in 𝒢j. We recall that nij is the order of ζij.We have the following description.Lemma 2.1.
Fix[A,B] a compact interval of ℝ.
There exists a finite numberp of real extrema of W in [A,B].    (i)
Ifp=0, there exists Y>0 such that(2.3)(E0-W)-1(ℝ)∩{ζ∈SY;Reζ∈[A,B]}=[A,B].
(ii)
For p>0, one denotes by {ζ1,…,ζp} the real extrema of W in [A,B]. There exists Y>0 and a sequence {Σi1,…,Σini-1}i∈{1…p} of disjoint and strictly vertical lines of ℂ+ starting at ζi such that(2.4)(E0-W)-1(ℝ)∩{ζ∈SY;Reζ∈[A,B]}=[A,B]⋃i=1p(⋃k=1ni-1(Σik∪Σik¯)).


## 2.1.1. The SetW-1(ℝ)

AsE∈ℝ, the set (E-W)-1(ℝ) coincides with W-1(ℝ). It is 2π-periodic. 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 ofW of order ni on the real line, say ζi. Near ζi, the set W-1(ℝ) consists of a real segment and of ni-1 complex curves symmetric with respect to the real axis and intersecting the real axis only on ζi. The angle between two neighboring curves is equal to π/ni. Let Y>0. We set SY={-Y≤Imζ≤Y}. We assume that Y is so small that  (i)
SY is contained in the domain of analyticity of W;(ii)
the setW-1(ℝ)∩SY consists of the real line and of the complex lines passing through the real extrema of W.An example of subsetW-1(ℝ) is shown in Figure 3.Figure 3
A subset ofW-1(ℝ)∩SY.Figure 4
Bands, gaps, and the action of mapping of the quasimomentumkp on some curves.


## 2.1.2. Notations and Description of(E-W)-1(σ(H0))

For allE∈J, we write (2.1)(E-W)-1(σ(H0))∩ℝ=⋃k∈ℤ⋃j=1N{[φj-(E),φj+(E)]+2kπ},
with the following properties.  (i)
φ1-(E)<φ1+(E)<φ2-(E)<⋯<φN-(E)<φN+(E),0<φN+(E)-φ1-(E)<2π.(ii)
As we deal with the case when the iso-energy curve has a real branch that is extended along the momentum direction, without loss of generality we consider that the connected component[φ1-(E),φ1+(E)] is associated to a connected component of Γℝ(E) which is an unbounded vertical curve. See Remark 2.3.(iii)
We generally define the following(2.2)φj+k+N-(E)=φj-(E)+2kπ,∀j∈{1,…,N},∀k∈ℤ.
We set ℬj(E) = [φj-(E),φj+(E)],𝒵 = ∪j=1Nℬj(E),𝒢j(E) = ]φj+(E),φj+1-(E)[, and 𝒢N(E) =  (φN+(E),φ1-(E)+2π) = (φN+(E),2π)∪(2π,2π+φ1-(E)). Let (ζij)1≤i≤pj be the extrema of W in 𝒢j. We recall that nij is the order of ζij.We have the following description.Lemma 2.1.
Fix[A,B] a compact interval of ℝ.
There exists a finite numberp of real extrema of W in [A,B].    (i)
Ifp=0, there exists Y>0 such that(2.3)(E0-W)-1(ℝ)∩{ζ∈SY;Reζ∈[A,B]}=[A,B].
(ii)
For p>0, one denotes by {ζ1,…,ζp} the real extrema of W in [A,B]. There exists Y>0 and a sequence {Σi1,…,Σini-1}i∈{1…p} of disjoint and strictly vertical lines of ℂ+ starting at ζi such that(2.4)(E0-W)-1(ℝ)∩{ζ∈SY;Reζ∈[A,B]}=[A,B]⋃i=1p(⋃k=1ni-1(Σik∪Σik¯)).


## 2.2. Bloch Solutions

Letψ be a solution of the equation(2.5)-d2dx2ψ(x,ℰ)+V(x)ψ(x,ℰ)=ℰψ(x,ℰ),x∈ℝ
satisfying the relation(2.6)ψ(x+1,ℰ)=λ(ℰ)ψ(x,ℰ)
for all x∈ℝ and some nonvanishing 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 Section1.1.2, we denote the spectral bands of the periodic Schrödinger operator by [E2n+1,E2n+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.One can construct a Bloch solutionψ(x,ℰ) meromorphic on 𝒮. It is normalized by the condition ψ(1,ℰ)≡1. The poles of this solution are located in the open spectral gaps or at their edges; the closure of each spectral gap contains exactly one pole that, moreover, is simple. It is located either on 𝒮+ or on 𝒮-. The position of the pole is independent of x.Forℰ∈𝒮, we denote by ℰ̂ the point on 𝒮 different from ℰ and having the same projection on ℂ as ℰ. We let (2.7)ψ̂(x,ℰ)=ψ(x,ℰ̂),ℰ̂∈𝒮.
The function ψ̂(x,ℰ) is another Bloch solution of (2.5). Except at the edges of the spectrum, the functions ψ and ψ̂ are linearly independent solutions of (2.5). In the spectral gaps, ψ and ψ̂ are real-valued functions of x, and, on the spectral bands, they differ only by complex conjugation.


## 2.3. The Bloch Quasimomentum

Consider the Bloch solutionψ(x,ℰ). The corresponding Floquet multiplier λ(ℰ) is analytic on 𝒮. Represent it in the form λ(ℰ)=exp(ik(ℰ)). The function ℰ↦k(ℰ) is the Bloch Quasimomentum of H0. Its inverse k↦E(k) is the dispersion  relation of H0. A branching point ζ is a point where k′(ζ)=0.LetD be a simply connected domain containing no branch point of the Bloch Quasimomentum. In D, one can fix an analytic single-valued branch of k, say k0. All the other single-valued branches of k that are analytic in D are related to k0 by the following formulae:
(2.8)k(ℰ)=±k0(ℰ)+2πl,l∈ℤ.
Consider ℂ+, the upper half plane of the complex plane. On ℂ+, one can fix a single-valued analytic branch of the Quasimomentum continuous up to the real line. It can be determined uniquely by the conditions Rek(ℰ+i0)=0 and Imk(ℰ+i0)>0 for ℰ<E1. We call this branch the main branch of the Bloch Quasimomentum and denote it by kp.The functionkp conformally maps ℂ+ onto the first quadrant of the complex plane cut at compact vertical slits starting at the points πl,l∈ℕ. It is monotonically increasing along the spectral zones so that [E2n-1,E2n], the nth spectral band, is mapped on the interval [π(n-1),πn]. Along any open gap, Rekp(ℰ+i0) is constant, and Imkp(ℰ+i0) is positive and has only one nondegenerate maximum.Considerℂ0, the complex plane cut along the spectral the real line from E1 to +∞. In Figure 4, we drew two curves in ℂ0 and their images under the transformation ℰ↦kp(ℰ).All the branch points ofkp are of square root type. Let El be a branch point of kp. In a sufficiently small neighborhood of El, the function kp is analytic in E-El, and(2.9)kp(ℰ)-kp(El)=clℰ-El+o(ℰ-El),cl≠0.
Finally, we note that the main branch can be continued analytically to the complex plane cut along (-∞,E1] and the spectral gaps ]E2n,E2n+1[,n∈ℕ*, of the periodic operator H0.


## 2.4. A Meromorphic Function

Now let us discuss a function playing an important role in the adiabatic constructions.In [10], it is shown that, on 𝒮, there is a meromorphic function ω having the following properties:  (i)
the differentialΩ=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′;(ii)
all the poles ofΩ are simple;(iii)
if the residue ofΩ at a point p is denoted by respΩ, one has
(2.10)respΩ=1,∀p∈P∖Q,resqΩ=-12,∀q∈Q∖P,resrΩ=12,∀r∈P∩Q.(iv)
ifℰ∈𝒮 projects into a gap, then ω(ℰ)∈ℝ.(v)
ifℰ∈𝒮 projects inside a band, then ω(ℰ)¯=ω(ℰ̂).


### 2.4.1. The Complex Momentum

It is the main analytic object of the complex WKB method. Letζ∈SY. We define κ, in 𝒟(W) the domain of analyticity of W, by(2.11)κ(ζ)=k(E-W(ζ)).
Here, k is the Bloch Quasimomentum defined in Section 2.3. Though κ depends on E, we omit the E-dependence. Relation (2.11) translates the properties of k into properties of κ. Hence, ζ↦κ(ζ) is a multivalued analytic function, and its branch points are related to the branch points of the Quasimomentum by the relations(2.12)E-W(ζ)=El,l=1,2,3,….
Let ζ0 be a branch point of κ. If W'(ζ0)≠0, then ζ0 is a branch point of square root type.IfD⊂𝒟(W) is a simply connected set containing no branch points of κ, we call it regular. Let κp 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:(2.13)κm±=±κp+2πm.
Here ± and m∈ℤ are indexing the branches.


### 2.4.2. Index of an Interval[φj-(E),φj+(E)]

Fixj∈{1,…,N}. Fix a continuous branch κj of the complex momentum on [φj-(E),φj+(E)]. We define(2.14)κj+=κj(φj+);κj-=κj(φj-);pjπ=κj+-κj-.pj is the index of [φj-(E),φj+(E)] associated to κj.Let us give some properties ofpj.Lemma 2.2.
Assume that(H4) is satisfied. The indices pj have the following properties.  (1)
Forj∈{1,…,N},pj∈{-1,0,1}.(2)
∑j=1N|pj|∈2ℕ.Proof.
The points(E-W)(φj-(E)) and (E-W)(φj+(E)) are the ends of a band of σ(H0): they are distinct or coincide. If they coincide, that is, if (E-W)(φj-(E))=(E-W)(φj+(E)), the index pj satisfies pj=0. Else, we consider kj the branch of the Quasimomentum associated to κj. We have that
(2.15)|kj((E-W)(φj-(E)))-kj((E-W)(φj+(E)))|=π,
see (2.8), and |pj|=1.
Let us prove point(2).As for any j∈{1,…,N}, we have |pj|=pjmod2.  We write
(2.16)∑j=1N|pj|≡∑j=1Npjmod2,∑j=1Npj=∑j=1Nκj(φj+)-κj(φj-)π=κN(φN+)-κ1(φ1-)π+∑j=1N-1κj+1(φj+1-)-κj(φj+)π.
For j∈{1,…,N-1},(E-W)(φj+1-(E)) and (E-W)(φj+(E)) are the ends of a same gap and
(2.17)kj+1((E-W)(φj+1-(E)))≡kj((E-W)(φj+(E)))[2π].
By periodicity, (E-W)(φ1-(E)) and (E-W)(φN+(E)) are the ends of the same gap.
This ends the proof of Lemma2.2.Ifpj≠0, we say that we cross a band. In this case, (E-W)(φj-)≠(E-W)(φj+) and the associated connected component of the iso-energy curve is unbounded vertically.We notice thatp1≠0, and thus ∑j=1N|pj|>0.Remark 2.3.
We can choose the determination ofκ such that p1=1.


### 2.4.3. Tunneling Coefficients

Forj∈{1,…,N}, we denote by γj a smooth closed curve that goes once around [φj+(E),φj+1-(E)]. 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 Sj given by(2.18)Sj(E)=i∮γjκ(ζ)dζ,∀j∈{1,…N}.
It is straightforward to prove that for E∈J, each of these actions is real and nonzero and that Sj(E) is analytic in a complex neighborhood of J (for analogous statements, we refer to [13, 14]). By definition, we choose the direction of the integration so that all the tunneling actions is positive. We set (2.19)tj=e-(1/2ε)Sj(E).Sj(E) is called the tunneling action; we choose the branch κ so that on J,Sj(E) is positive. We define (2.20)T(E,ε)=∏j=1Ntj.
So we get
(2.21)T(E,ε)=e-∑j=1NSj(E)/2ε.Sj(E) being positive we get that T is exponentially small. For more details on the properties of tunneling coefficients, see [5, Section  10].


## 2.4.1. The Complex Momentum

It is the main analytic object of the complex WKB method. Letζ∈SY. We define κ, in 𝒟(W) the domain of analyticity of W, by(2.11)κ(ζ)=k(E-W(ζ)).
Here, k is the Bloch Quasimomentum defined in Section 2.3. Though κ depends on E, we omit the E-dependence. Relation (2.11) translates the properties of k into properties of κ. Hence, ζ↦κ(ζ) is a multivalued analytic function, and its branch points are related to the branch points of the Quasimomentum by the relations(2.12)E-W(ζ)=El,l=1,2,3,….
Let ζ0 be a branch point of κ. If W'(ζ0)≠0, then ζ0 is a branch point of square root type.IfD⊂𝒟(W) is a simply connected set containing no branch points of κ, we call it regular. Let κp 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:(2.13)κm±=±κp+2πm.
Here ± and m∈ℤ are indexing the branches.


## 2.4.2. Index of an Interval[φj-(E),φj+(E)]

Fixj∈{1,…,N}. Fix a continuous branch κj of the complex momentum on [φj-(E),φj+(E)]. We define(2.14)κj+=κj(φj+);κj-=κj(φj-);pjπ=κj+-κj-.pj is the index of [φj-(E),φj+(E)] associated to κj.Let us give some properties ofpj.Lemma 2.2.
Assume that(H4) is satisfied. The indices pj have the following properties.  (1)
Forj∈{1,…,N},pj∈{-1,0,1}.(2)
∑j=1N|pj|∈2ℕ.Proof.
The points(E-W)(φj-(E)) and (E-W)(φj+(E)) are the ends of a band of σ(H0): they are distinct or coincide. If they coincide, that is, if (E-W)(φj-(E))=(E-W)(φj+(E)), the index pj satisfies pj=0. Else, we consider kj the branch of the Quasimomentum associated to κj. We have that
(2.15)|kj((E-W)(φj-(E)))-kj((E-W)(φj+(E)))|=π,
see (2.8), and |pj|=1.
Let us prove point(2).As for any j∈{1,…,N}, we have |pj|=pjmod2.  We write
(2.16)∑j=1N|pj|≡∑j=1Npjmod2,∑j=1Npj=∑j=1Nκj(φj+)-κj(φj-)π=κN(φN+)-κ1(φ1-)π+∑j=1N-1κj+1(φj+1-)-κj(φj+)π.
For j∈{1,…,N-1},(E-W)(φj+1-(E)) and (E-W)(φj+(E)) are the ends of a same gap and
(2.17)kj+1((E-W)(φj+1-(E)))≡kj((E-W)(φj+(E)))[2π].
By periodicity, (E-W)(φ1-(E)) and (E-W)(φN+(E)) are the ends of the same gap.
This ends the proof of Lemma2.2.Ifpj≠0, we say that we cross a band. In this case, (E-W)(φj-)≠(E-W)(φj+) and the associated connected component of the iso-energy curve is unbounded vertically.We notice thatp1≠0, and thus ∑j=1N|pj|>0.Remark 2.3.
We can choose the determination ofκ such that p1=1.


## 2.4.3. Tunneling Coefficients

Forj∈{1,…,N}, we denote by γj a smooth closed curve that goes once around [φj+(E),φj+1-(E)]. 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 Sj given by(2.18)Sj(E)=i∮γjκ(ζ)dζ,∀j∈{1,…N}.
It is straightforward to prove that for E∈J, each of these actions is real and nonzero and that Sj(E) is analytic in a complex neighborhood of J (for analogous statements, we refer to [13, 14]). By definition, we choose the direction of the integration so that all the tunneling actions is positive. We set (2.19)tj=e-(1/2ε)Sj(E).Sj(E) is called the tunneling action; we choose the branch κ so that on J,Sj(E) is positive. We define (2.20)T(E,ε)=∏j=1Ntj.
So we get
(2.21)T(E,ε)=e-∑j=1NSj(E)/2ε.Sj(E) being positive we get that T is exponentially small. For more details on the properties of tunneling coefficients, see [5, Section  10].


## 3. The Proof of Theorem1.2

### 3.1. The Asymptotics of the Lyapunov Exponent

#### 3.1.1. Spectral Results

One of the main objects of the spectral theory of quasi-periodic operators is the Lyapunov exponent (for a definition and additional information, see, e.g., [9]). The main result of this section is as follows.Theorem 3.1.
We assume that the assumptions(H1)–(H4) are satisfied. Then, on the interval J, for sufficiently small irrational ε/2π, the Lyapunov exponent Θ(E,ε) of (1.1) is positive and satisfies the asymptotics
(3.1)Θ(E,ε)=ε2π∑j=1Nln1tj+o(1)=14π∑j=1NSj(E)+o(1).This theorem implies that ifε/2π is sufficiently small and irrational, then, the Lyapunov exponent is positive for all E∈J.


### 3.2. The Monodromy Matrix and the Lyapunov Exponents

The main object of our study in this subsection is the monodromy matrix for the family of (1.1), and we define it briefly (we refer the reader to [5, 13]). In this paper, we compute the asymptotics of its Fourier expansion in the adiabatic limit.


#### 3.2.1. Definition of the Monodromy Matrix

FixE∈ℝ. Consider the family of differential equations indexed by z∈ℝ:(3.2)(-d2dx2+V(x-z)+W(ɛx))ψ(x)=Eψ(x).Definition 3.2.
We say that(ψi)i∈{1,2} is a consistent basis of solutions to (3.2) if the two functions ((x,z)↦ψi(x,z,E))i∈{1,2} are a basis of solutions to (3.2) whose Wronskian is independent of z and that are 1-periodic in z, that is, that satisfy
(3.3)∀x∈ℝ,∀z∈ℝ,∀i∈{1,2},ψi(x,z+1,E)=ψi(x,z,E).We refer the reader to [5, 10] about the existence and details on consistent basis of solutions to (3.2).The functions((x,z)↦ψi(x+2π/ε,z+2π/ε,E))i∈{1,2} being also solutions of (3.2), we get the relation(3.4)Ψ(x+2πε,z+2πε,E)=M(z,E)Ψ(x,z,E),
where  (i)
Ψ(x,z,E)=(ψ1(x,z,E)ψ2(x,z,E)),(ii)
M(z,E) is a 2×2-matrix with coefficients independent of x.The matrixM is called themonodromy matrix associated to the consistent basis (ψ1,2). We recall the following properties of this matrix:(3.5)detM(z,E)≡1,M(z+1,E)=M(z,E),∀z∈ℝ.
The Matrix M belongs to SL(2,ℝ) which is known to be isomorph to SU(1,1).


### 3.3. The Lyapunov Exponents and the Monodromy Equation

Consider now a 1-periodic,SL(2,ℂ)-valued function, say, z↦M̃, and h>0 irrational. Consider the finite difference equation:(3.6)Fn+1=M̃(z+nh)Fn∀n∈ℤ,Fn∈ℂ2.
Going from (1.1) to the (3.6) is close to the monodromization transformation introduced in [8] to construct Bloch solutions of difference equation. Indeed, it appears that the behavior of solutions of (1.1) for x→∓∞ repeats the behavior of solutions of the monodromy equation for n→∓∞. And it is a well-known fact that the spectral properties of the one-dimensional Schrödinger equations can be described in terms of the behavior of its solutions as x→∓∞.The Lyapunov exponent of the finite difference equation (3.6) is(3.7)θ(M̃,h)=limN↦+∞1Nlog∥PN(z,h)∥,
where the matrix cocycle (PN(z,h))N∈ℕ is defined as(3.8)PN(z,h)=M̃(z+Nh)·M̃(z+(N-1)h)⋯M̃(z+h)·M̃(z).
It is well known that if h is irrational, and M̃ is sufficiently regular in z, then the limit (3.7) exists for almost all z and is independent of z.Seth≡2π/ε [2]. Let M be the monodromy matrix associated to a consistent basis (ψ1,2). Consider the monodromy equation:(3.9)Fn+1=M(z+nh,E)Fn∀n∈ℤ,Fn∈ℂ2.
The Lyapunov exponent of the monodromy equation (3.9) is defined by (3.10)θ(E,ɛ)=θ(M(z,E),h).
There are several deep relations between (3.2) and the monodromy equation (3.9) (see [5, 15]). We describe only one of them. Recall that Θ(E,ε) is the Lyapunov exponent of (1.1). We have the following result.Theorem 3.3 (see [5]).
Assume thatε/2π is irrational. The Lyapunov exponents Θ(E,ε) and θ(E,ε) are related by the following relation:
(3.11)Θ(E,ε)=ε2πθ(E,ɛ).


### 3.4. The Asymptotics of the Monodromy Matrix

AsW and V are real on the real line, we construct a monodromy matrix of the following form:(3.12)M(z,E)=(A(z,E)B(z,E)B(z¯,E¯)¯A(z¯,E¯)¯).
To get (3.12), it suffices to consider a basis of solutions of the form (u;u¯). The details on the existence and the construction of such a basis are developed in [13].The following result gives the asymptotics ofA and B in the adiabatic case.Theorem 3.4.
LetE0 be in J. There exists Y>0 and V0, a neighborhood of E0, such that, for sufficiently small ε, the family of (3.2) has a consistent basis of solutions for which the corresponding monodromy matrix M is analytic in (z,E)∈{z∈ℂ;|Imz|<Y/ε}×V0 and has the form (3.12). When ε tends to 0, the coefficients A and B admit the asymptotics
(3.13)A=A+(E,ε)e-2iQ+πz[1+o(1)],B=B+(E,ε)e-2iP+πz[1+o(1)],0<Imz<Yε,(3.14)A=A-(E,ε)e2iQ-πz[1+o(1)],B=B-(E,ε)e2iP-πz[1+o(1)],-Yε<Imz<0.
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∈V0∩ℝ, one has
(3.15)1C<T(E,ε)|A±(E,ε)|<C,1C<T(E,ε)|B±(E,ε)|<C,
where T(E,ε) is defined in (2.21).
ForY1 and Y2 such that 0<Y1<Y2<Y, there exists V=V(Y1,Y2) a neighborhood of E0 such that the asymptotics (3.13) and (3.14) for A and B are uniform in (z,E)∈{z∈ℂ;Y1/ε<|Imz|<Y2/ε}×V.Remark 3.5.
The coefficientsA+, A-,B+, and B- are the leading terms of the asymptotics of the Q∓th and P∓th Fourier coefficients of the monodromy matrix coefficients. From Theorem 3.4, one deduces that, in the strip {-Y<Imζ<Y}, only a few Fourier series terms of the monodromy matrix dominate.


### 3.5. The Proof of Theorem3.1

#### 3.5.1. The Upper Bound

FixP=max|P∓|,|Q∓|. The asymptotics (3.13) and (3.14) and estimates (3.15) imply the following estimates for the coefficients of M(z,E), the monodromy matrix:(3.16)|A|,|B|≤C(y0)·T(E)-1e2πPy0/ε,Imz=y0ε,|A|,|B|≤C(y0)·T(E)-1e2πPy0/ε,Imz=-y0ε.
Here, C(y0) is a positive constant independent of ε,Rez, and E. The estimates are valid for sufficiently small ε. We recall that M is analytic and 1-periodic in z. Equation (3.16) and the maximum principle imply that (3.17)|A|,|B|≤2C(y0)T(E)-1exp(2πPy0ε),z∈ℝ.
This leads to the following upper bound for the Lyapunov exponent for the matrix cocycle generated by M(z,E):(3.18)θ(E,ε)≤log(T(E)-1)+C+2πPy0ε,
where C is a constant independent of E and ε. Using (3.11) one gets(3.19)Θ(E,ε)≤ε2πlog(T(E)-1)+εC+2πPy0.


#### 3.5.2. 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 [8].Proposition 3.6.
Fixε0>0. Assume that there exist y0 and y1 such that 0<y0<y1<∞ and such that, for any ε∈(0,ε0), one has  (i)
the functionz→M(z,ε) is analytic in the strip S={z∈ℂ;0≤Imz≤y1/ε};(ii)
in the stripS1={z∈ℂ;y0/ε≤Imz≤y1/ε}⊂S,M(z,ε) admits the representation
(3.20)M(z,ε)=λ(ε)ei2πn0z·(M0(ε)+M1(z,ε));
for some constant λ(ε), some integer n0, and a matrix M0(ε), all of them independent of z;(iii)
M(z,E)=(10β(ε)α(ε));(iv)
there exist constantsβ>0 and α∈(0,1) independent of ε such that |α(ε)|≤α and |β(ε)|≤β;(v)
m(ε)=supz∈S1∥M1(z,ε)∥→0 as ε→0. Then, there exists C>0 and ɛ1>0 (both depending only on y0,y1,α,β, and ε↦m(ε)) such that, if 0<ε<ε1, one has
(3.21)θ(M(·,ε),h)>log|λ(ε)|-Cm(ε).Proposition3.6 is used by applying the arguments of [5, 10] to get the lower bound for the Lyapunov exponent.First forσ=(0110) we prove that the matrix σM(z,E)σ completes the assumption of Proposition 3.6.Lety0 and y1 be fixed such that 0<y0<y1<Y. The asymptotics of the monodromy matrix are uniform for z in S={z∈ℂ;y0/ε≤Imz≤y1/ε} and E∈V0.Let us assume thatn=NN in (5.16) is even; then the following relations hold:(3.22)Q-=Q++1,Q-=P-+1,Q-=P+.
Equation (3.22) is derived from the expression of Q± and P±, given in Section 5.3. λj±,βj± are equals for 2≤j≤N. The first expression of formula (3.22) deduced the relation between β1+ and β1- (see (5.27) and (5.32)), the second one from the relation between λ1- and β1- (see (5.21) and (5.32)), and the last one from the relation between λ1+ and β1- (see (5.17) and (5.32)). We notice that (3.22) depends only on the indices of 𝒢1 because we constructed the consistent basis near [φ1-(E),φ1+(E)]. See Section 4.4.ForE∈V0∩ℝ and z∈S, Theorem 3.4 implies that (3.23)A(z¯,E)¯=A-(E,ε)¯e-2πiQ-z(1+o(1)),A(z,E)A(z¯,E)¯=o(1),B(z¯,E)¯A(z¯,E)¯=o(1),B(z,E)A(z¯,E)¯=c(E)(1+o(1)),
where c(E) is independent of z and bounded by a constant uniformly in ε and E. So, we have (3.24)σ·M(z,E)·σ=A-(E,ε)¯e-2πiQ-z·((10c(E)0)+o(1)).
This gives that the matrix-valued function (3.25)z↦σ·M(z,E)·σ
satisfies the assumptions of Proposition 3.6.Remark 3.7.
Whenn=NN in (5.16) is odd, then we get
(3.26)Q+=Q-+1,P-=Q+,P++1=Q+.
For E∈V0∩ℝ and z∈S, Theorem 3.4 implies that
(3.27)A(z,E)=A+(E,ε)e-2πiQ+z(1+o(1)),A(z¯,E)¯A(z,E)=o(1),B(z¯,E)¯A(z¯,E)=c(E)(1+o(1)),B(z,E)A(z,E)=o(1),
where c(E) is independent of z and bounded by a constant uniformly in ε and E. So, we have
(3.28)σ·M(z,E)·σ=A+(E,ε)e-2πiQ+z·((0c(E)01)+o(1)).
The most important properties of the matrix M(z,E) in Proposition 3.6 are that the bigger eigenvalue is 1 [8].Using (3.21), we get that the Lyapunov exponent θ(E,ε) of the matrix cocycle associated to (M(·,E),h) satisfies the estimates(3.29)θ(E,ε)≥log|A-|+o(1).
Taking into account (1.2) we get that (3.30)Θ(E,ε)≥ε2πlog|A-|+o(ε).
By (3.15) we get(3.31)Θ(E,ε)≥ε2πlog(T(E)-1)+O(ε).


#### 3.5.3. Conclusion

Now we obtain (3.1), by comparing (3.19) and (3.31). Indeed we see that(3.32)Θ(E,ε)=ε2πlog(T(E)-1)+o(1).
The expression of T(E) given by (3.32) and (3.31) gives (3.1) for any E∈V0∩ℝ.Recall thatV0∩ℝ is an open interval containing E0∈J. The above construction can be carried out for any E0∈J. The end of the proof of Theorem 1.2 follows from the compactness of the interval J.


## 3.1. The Asymptotics of the Lyapunov Exponent

### 3.1.1. Spectral Results

One of the main objects of the spectral theory of quasi-periodic operators is the Lyapunov exponent (for a definition and additional information, see, e.g., [9]). The main result of this section is as follows.Theorem 3.1.
We assume that the assumptions(H1)–(H4) are satisfied. Then, on the interval J, for sufficiently small irrational ε/2π, the Lyapunov exponent Θ(E,ε) of (1.1) is positive and satisfies the asymptotics
(3.1)Θ(E,ε)=ε2π∑j=1Nln1tj+o(1)=14π∑j=1NSj(E)+o(1).This theorem implies that ifε/2π is sufficiently small and irrational, then, the Lyapunov exponent is positive for all E∈J.


## 3.1.1. Spectral Results

One of the main objects of the spectral theory of quasi-periodic operators is the Lyapunov exponent (for a definition and additional information, see, e.g., [9]). The main result of this section is as follows.Theorem 3.1.
We assume that the assumptions(H1)–(H4) are satisfied. Then, on the interval J, for sufficiently small irrational ε/2π, the Lyapunov exponent Θ(E,ε) of (1.1) is positive and satisfies the asymptotics
(3.1)Θ(E,ε)=ε2π∑j=1Nln1tj+o(1)=14π∑j=1NSj(E)+o(1).This theorem implies that ifε/2π is sufficiently small and irrational, then, the Lyapunov exponent is positive for all E∈J.


## 3.2. The Monodromy Matrix and the Lyapunov Exponents

The main object of our study in this subsection is the monodromy matrix for the family of (1.1), and we define it briefly (we refer the reader to [5, 13]). In this paper, we compute the asymptotics of its Fourier expansion in the adiabatic limit.


### 3.2.1. Definition of the Monodromy Matrix

FixE∈ℝ. Consider the family of differential equations indexed by z∈ℝ:(3.2)(-d2dx2+V(x-z)+W(ɛx))ψ(x)=Eψ(x).Definition 3.2.
We say that(ψi)i∈{1,2} is a consistent basis of solutions to (3.2) if the two functions ((x,z)↦ψi(x,z,E))i∈{1,2} are a basis of solutions to (3.2) whose Wronskian is independent of z and that are 1-periodic in z, that is, that satisfy
(3.3)∀x∈ℝ,∀z∈ℝ,∀i∈{1,2},ψi(x,z+1,E)=ψi(x,z,E).We refer the reader to [5, 10] about the existence and details on consistent basis of solutions to (3.2).The functions((x,z)↦ψi(x+2π/ε,z+2π/ε,E))i∈{1,2} being also solutions of (3.2), we get the relation(3.4)Ψ(x+2πε,z+2πε,E)=M(z,E)Ψ(x,z,E),
where  (i)
Ψ(x,z,E)=(ψ1(x,z,E)ψ2(x,z,E)),(ii)
M(z,E) is a 2×2-matrix with coefficients independent of x.The matrixM is called themonodromy matrix associated to the consistent basis (ψ1,2). We recall the following properties of this matrix:(3.5)detM(z,E)≡1,M(z+1,E)=M(z,E),∀z∈ℝ.
The Matrix M belongs to SL(2,ℝ) which is known to be isomorph to SU(1,1).


## 3.2.1. Definition of the Monodromy Matrix

FixE∈ℝ. Consider the family of differential equations indexed by z∈ℝ:(3.2)(-d2dx2+V(x-z)+W(ɛx))ψ(x)=Eψ(x).Definition 3.2.
We say that(ψi)i∈{1,2} is a consistent basis of solutions to (3.2) if the two functions ((x,z)↦ψi(x,z,E))i∈{1,2} are a basis of solutions to (3.2) whose Wronskian is independent of z and that are 1-periodic in z, that is, that satisfy
(3.3)∀x∈ℝ,∀z∈ℝ,∀i∈{1,2},ψi(x,z+1,E)=ψi(x,z,E).We refer the reader to [5, 10] about the existence and details on consistent basis of solutions to (3.2).The functions((x,z)↦ψi(x+2π/ε,z+2π/ε,E))i∈{1,2} being also solutions of (3.2), we get the relation(3.4)Ψ(x+2πε,z+2πε,E)=M(z,E)Ψ(x,z,E),
where  (i)
Ψ(x,z,E)=(ψ1(x,z,E)ψ2(x,z,E)),(ii)
M(z,E) is a 2×2-matrix with coefficients independent of x.The matrixM is called themonodromy matrix associated to the consistent basis (ψ1,2). We recall the following properties of this matrix:(3.5)detM(z,E)≡1,M(z+1,E)=M(z,E),∀z∈ℝ.
The Matrix M belongs to SL(2,ℝ) which is known to be isomorph to SU(1,1).


## 3.3. The Lyapunov Exponents and the Monodromy Equation

Consider now a 1-periodic,SL(2,ℂ)-valued function, say, z↦M̃, and h>0 irrational. Consider the finite difference equation:(3.6)Fn+1=M̃(z+nh)Fn∀n∈ℤ,Fn∈ℂ2.
Going from (1.1) to the (3.6) is close to the monodromization transformation introduced in [8] to construct Bloch solutions of difference equation. Indeed, it appears that the behavior of solutions of (1.1) for x→∓∞ repeats the behavior of solutions of the monodromy equation for n→∓∞. And it is a well-known fact that the spectral properties of the one-dimensional Schrödinger equations can be described in terms of the behavior of its solutions as x→∓∞.The Lyapunov exponent of the finite difference equation (3.6) is(3.7)θ(M̃,h)=limN↦+∞1Nlog∥PN(z,h)∥,
where the matrix cocycle (PN(z,h))N∈ℕ is defined as(3.8)PN(z,h)=M̃(z+Nh)·M̃(z+(N-1)h)⋯M̃(z+h)·M̃(z).
It is well known that if h is irrational, and M̃ is sufficiently regular in z, then the limit (3.7) exists for almost all z and is independent of z.Seth≡2π/ε [2]. Let M be the monodromy matrix associated to a consistent basis (ψ1,2). Consider the monodromy equation:(3.9)Fn+1=M(z+nh,E)Fn∀n∈ℤ,Fn∈ℂ2.
The Lyapunov exponent of the monodromy equation (3.9) is defined by (3.10)θ(E,ɛ)=θ(M(z,E),h).
There are several deep relations between (3.2) and the monodromy equation (3.9) (see [5, 15]). We describe only one of them. Recall that Θ(E,ε) is the Lyapunov exponent of (1.1). We have the following result.Theorem 3.3 (see [5]).
Assume thatε/2π is irrational. The Lyapunov exponents Θ(E,ε) and θ(E,ε) are related by the following relation:
(3.11)Θ(E,ε)=ε2πθ(E,ɛ).


## 3.4. The Asymptotics of the Monodromy Matrix

AsW and V are real on the real line, we construct a monodromy matrix of the following form:(3.12)M(z,E)=(A(z,E)B(z,E)B(z¯,E¯)¯A(z¯,E¯)¯).
To get (3.12), it suffices to consider a basis of solutions of the form (u;u¯). The details on the existence and the construction of such a basis are developed in [13].The following result gives the asymptotics ofA and B in the adiabatic case.Theorem 3.4.
LetE0 be in J. There exists Y>0 and V0, a neighborhood of E0, such that, for sufficiently small ε, the family of (3.2) has a consistent basis of solutions for which the corresponding monodromy matrix M is analytic in (z,E)∈{z∈ℂ;|Imz|<Y/ε}×V0 and has the form (3.12). When ε tends to 0, the coefficients A and B admit the asymptotics
(3.13)A=A+(E,ε)e-2iQ+πz[1+o(1)],B=B+(E,ε)e-2iP+πz[1+o(1)],0<Imz<Yε,(3.14)A=A-(E,ε)e2iQ-πz[1+o(1)],B=B-(E,ε)e2iP-πz[1+o(1)],-Yε<Imz<0.
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∈V0∩ℝ, one has
(3.15)1C<T(E,ε)|A±(E,ε)|<C,1C<T(E,ε)|B±(E,ε)|<C,
where T(E,ε) is defined in (2.21).
ForY1 and Y2 such that 0<Y1<Y2<Y, there exists V=V(Y1,Y2) a neighborhood of E0 such that the asymptotics (3.13) and (3.14) for A and B are uniform in (z,E)∈{z∈ℂ;Y1/ε<|Imz|<Y2/ε}×V.Remark 3.5.
The coefficientsA+, A-,B+, and B- are the leading terms of the asymptotics of the Q∓th and P∓th Fourier coefficients of the monodromy matrix coefficients. From Theorem 3.4, one deduces that, in the strip {-Y<Imζ<Y}, only a few Fourier series terms of the monodromy matrix dominate.


## 3.5. The Proof of Theorem3.1

### 3.5.1. The Upper Bound

FixP=max|P∓|,|Q∓|. The asymptotics (3.13) and (3.14) and estimates (3.15) imply the following estimates for the coefficients of M(z,E), the monodromy matrix:(3.16)|A|,|B|≤C(y0)·T(E)-1e2πPy0/ε,Imz=y0ε,|A|,|B|≤C(y0)·T(E)-1e2πPy0/ε,Imz=-y0ε.
Here, C(y0) is a positive constant independent of ε,Rez, and E. The estimates are valid for sufficiently small ε. We recall that M is analytic and 1-periodic in z. Equation (3.16) and the maximum principle imply that (3.17)|A|,|B|≤2C(y0)T(E)-1exp(2πPy0ε),z∈ℝ.
This leads to the following upper bound for the Lyapunov exponent for the matrix cocycle generated by M(z,E):(3.18)θ(E,ε)≤log(T(E)-1)+C+2πPy0ε,
where C is a constant independent of E and ε. Using (3.11) one gets(3.19)Θ(E,ε)≤ε2πlog(T(E)-1)+εC+2πPy0.


### 3.5.2. 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 [8].Proposition 3.6.
Fixε0>0. Assume that there exist y0 and y1 such that 0<y0<y1<∞ and such that, for any ε∈(0,ε0), one has  (i)
the functionz→M(z,ε) is analytic in the strip S={z∈ℂ;0≤Imz≤y1/ε};(ii)
in the stripS1={z∈ℂ;y0/ε≤Imz≤y1/ε}⊂S,M(z,ε) admits the representation
(3.20)M(z,ε)=λ(ε)ei2πn0z·(M0(ε)+M1(z,ε));
for some constant λ(ε), some integer n0, and a matrix M0(ε), all of them independent of z;(iii)
M(z,E)=(10β(ε)α(ε));(iv)
there exist constantsβ>0 and α∈(0,1) independent of ε such that |α(ε)|≤α and |β(ε)|≤β;(v)
m(ε)=supz∈S1∥M1(z,ε)∥→0 as ε→0. Then, there exists C>0 and ɛ1>0 (both depending only on y0,y1,α,β, and ε↦m(ε)) such that, if 0<ε<ε1, one has
(3.21)θ(M(·,ε),h)>log|λ(ε)|-Cm(ε).Proposition3.6 is used by applying the arguments of [5, 10] to get the lower bound for the Lyapunov exponent.First forσ=(0110) we prove that the matrix σM(z,E)σ completes the assumption of Proposition 3.6.Lety0 and y1 be fixed such that 0<y0<y1<Y. The asymptotics of the monodromy matrix are uniform for z in S={z∈ℂ;y0/ε≤Imz≤y1/ε} and E∈V0.Let us assume thatn=NN in (5.16) is even; then the following relations hold:(3.22)Q-=Q++1,Q-=P-+1,Q-=P+.
Equation (3.22) is derived from the expression of Q± and P±, given in Section 5.3. λj±,βj± are equals for 2≤j≤N. The first expression of formula (3.22) deduced the relation between β1+ and β1- (see (5.27) and (5.32)), the second one from the relation between λ1- and β1- (see (5.21) and (5.32)), and the last one from the relation between λ1+ and β1- (see (5.17) and (5.32)). We notice that (3.22) depends only on the indices of 𝒢1 because we constructed the consistent basis near [φ1-(E),φ1+(E)]. See Section 4.4.ForE∈V0∩ℝ and z∈S, Theorem 3.4 implies that (3.23)A(z¯,E)¯=A-(E,ε)¯e-2πiQ-z(1+o(1)),A(z,E)A(z¯,E)¯=o(1),B(z¯,E)¯A(z¯,E)¯=o(1),B(z,E)A(z¯,E)¯=c(E)(1+o(1)),
where c(E) is independent of z and bounded by a constant uniformly in ε and E. So, we have (3.24)σ·M(z,E)·σ=A-(E,ε)¯e-2πiQ-z·((10c(E)0)+o(1)).
This gives that the matrix-valued function (3.25)z↦σ·M(z,E)·σ
satisfies the assumptions of Proposition 3.6.Remark 3.7.
Whenn=NN in (5.16) is odd, then we get
(3.26)Q+=Q-+1,P-=Q+,P++1=Q+.
For E∈V0∩ℝ and z∈S, Theorem 3.4 implies that
(3.27)A(z,E)=A+(E,ε)e-2πiQ+z(1+o(1)),A(z¯,E)¯A(z,E)=o(1),B(z¯,E)¯A(z¯,E)=c(E)(1+o(1)),B(z,E)A(z,E)=o(1),
where c(E) is independent of z and bounded by a constant uniformly in ε and E. So, we have
(3.28)σ·M(z,E)·σ=A+(E,ε)e-2πiQ+z·((0c(E)01)+o(1)).
The most important properties of the matrix M(z,E) in Proposition 3.6 are that the bigger eigenvalue is 1 [8].Using (3.21), we get that the Lyapunov exponent θ(E,ε) of the matrix cocycle associated to (M(·,E),h) satisfies the estimates(3.29)θ(E,ε)≥log|A-|+o(1).
Taking into account (1.2) we get that (3.30)Θ(E,ε)≥ε2πlog|A-|+o(ε).
By (3.15) we get(3.31)Θ(E,ε)≥ε2πlog(T(E)-1)+O(ε).


### 3.5.3. Conclusion

Now we obtain (3.1), by comparing (3.19) and (3.31). Indeed we see that(3.32)Θ(E,ε)=ε2πlog(T(E)-1)+o(1).
The expression of T(E) given by (3.32) and (3.31) gives (3.1) for any E∈V0∩ℝ.Recall thatV0∩ℝ is an open interval containing E0∈J. The above construction can be carried out for any E0∈J. The end of the proof of Theorem 1.2 follows from the compactness of the interval J.


## 3.5.1. The Upper Bound

FixP=max|P∓|,|Q∓|. The asymptotics (3.13) and (3.14) and estimates (3.15) imply the following estimates for the coefficients of M(z,E), the monodromy matrix:(3.16)|A|,|B|≤C(y0)·T(E)-1e2πPy0/ε,Imz=y0ε,|A|,|B|≤C(y0)·T(E)-1e2πPy0/ε,Imz=-y0ε.
Here, C(y0) is a positive constant independent of ε,Rez, and E. The estimates are valid for sufficiently small ε. We recall that M is analytic and 1-periodic in z. Equation (3.16) and the maximum principle imply that (3.17)|A|,|B|≤2C(y0)T(E)-1exp(2πPy0ε),z∈ℝ.
This leads to the following upper bound for the Lyapunov exponent for the matrix cocycle generated by M(z,E):(3.18)θ(E,ε)≤log(T(E)-1)+C+2πPy0ε,
where C is a constant independent of E and ε. Using (3.11) one gets(3.19)Θ(E,ε)≤ε2πlog(T(E)-1)+εC+2πPy0.


## 3.5.2. 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 [8].Proposition 3.6.
Fixε0>0. Assume that there exist y0 and y1 such that 0<y0<y1<∞ and such that, for any ε∈(0,ε0), one has  (i)
the functionz→M(z,ε) is analytic in the strip S={z∈ℂ;0≤Imz≤y1/ε};(ii)
in the stripS1={z∈ℂ;y0/ε≤Imz≤y1/ε}⊂S,M(z,ε) admits the representation
(3.20)M(z,ε)=λ(ε)ei2πn0z·(M0(ε)+M1(z,ε));
for some constant λ(ε), some integer n0, and a matrix M0(ε), all of them independent of z;(iii)
M(z,E)=(10β(ε)α(ε));(iv)
there exist constantsβ>0 and α∈(0,1) independent of ε such that |α(ε)|≤α and |β(ε)|≤β;(v)
m(ε)=supz∈S1∥M1(z,ε)∥→0 as ε→0. Then, there exists C>0 and ɛ1>0 (both depending only on y0,y1,α,β, and ε↦m(ε)) such that, if 0<ε<ε1, one has
(3.21)θ(M(·,ε),h)>log|λ(ε)|-Cm(ε).Proposition3.6 is used by applying the arguments of [5, 10] to get the lower bound for the Lyapunov exponent.First forσ=(0110) we prove that the matrix σM(z,E)σ completes the assumption of Proposition 3.6.Lety0 and y1 be fixed such that 0<y0<y1<Y. The asymptotics of the monodromy matrix are uniform for z in S={z∈ℂ;y0/ε≤Imz≤y1/ε} and E∈V0.Let us assume thatn=NN in (5.16) is even; then the following relations hold:(3.22)Q-=Q++1,Q-=P-+1,Q-=P+.
Equation (3.22) is derived from the expression of Q± and P±, given in Section 5.3. λj±,βj± are equals for 2≤j≤N. The first expression of formula (3.22) deduced the relation between β1+ and β1- (see (5.27) and (5.32)), the second one from the relation between λ1- and β1- (see (5.21) and (5.32)), and the last one from the relation between λ1+ and β1- (see (5.17) and (5.32)). We notice that (3.22) depends only on the indices of 𝒢1 because we constructed the consistent basis near [φ1-(E),φ1+(E)]. See Section 4.4.ForE∈V0∩ℝ and z∈S, Theorem 3.4 implies that (3.23)A(z¯,E)¯=A-(E,ε)¯e-2πiQ-z(1+o(1)),A(z,E)A(z¯,E)¯=o(1),B(z¯,E)¯A(z¯,E)¯=o(1),B(z,E)A(z¯,E)¯=c(E)(1+o(1)),
where c(E) is independent of z and bounded by a constant uniformly in ε and E. So, we have (3.24)σ·M(z,E)·σ=A-(E,ε)¯e-2πiQ-z·((10c(E)0)+o(1)).
This gives that the matrix-valued function (3.25)z↦σ·M(z,E)·σ
satisfies the assumptions of Proposition 3.6.Remark 3.7.
Whenn=NN in (5.16) is odd, then we get
(3.26)Q+=Q-+1,P-=Q+,P++1=Q+.
For E∈V0∩ℝ and z∈S, Theorem 3.4 implies that
(3.27)A(z,E)=A+(E,ε)e-2πiQ+z(1+o(1)),A(z¯,E)¯A(z,E)=o(1),B(z¯,E)¯A(z¯,E)=c(E)(1+o(1)),B(z,E)A(z,E)=o(1),
where c(E) is independent of z and bounded by a constant uniformly in ε and E. So, we have
(3.28)σ·M(z,E)·σ=A+(E,ε)e-2πiQ+z·((0c(E)01)+o(1)).
The most important properties of the matrix M(z,E) in Proposition 3.6 are that the bigger eigenvalue is 1 [8].Using (3.21), we get that the Lyapunov exponent θ(E,ε) of the matrix cocycle associated to (M(·,E),h) satisfies the estimates(3.29)θ(E,ε)≥log|A-|+o(1).
Taking into account (1.2) we get that (3.30)Θ(E,ε)≥ε2πlog|A-|+o(ε).
By (3.15) we get(3.31)Θ(E,ε)≥ε2πlog(T(E)-1)+O(ε).


## 3.5.3. Conclusion

Now we obtain (3.1), by comparing (3.19) and (3.31). Indeed we see that(3.32)Θ(E,ε)=ε2πlog(T(E)-1)+o(1).
The expression of T(E) given by (3.32) and (3.31) gives (3.1) for any E∈V0∩ℝ.Recall thatV0∩ℝ is an open interval containing E0∈J. The above construction can be carried out for any E0∈J. The end of the proof of Theorem 1.2 follows from the compactness of the interval J.


## 4. The Complex WKB Method for Adiabatic Problems

In this section, following [10, 16, 17], we describe the complex WKB method for adiabatically perturbed periodic Schrödinger equations:(4.1)-d2dx2ψ(x)+[V(x)+W(εx+ζ)]ψ(x)=Eψ(x),x∈ℝ.
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 Lloc2 and that W is analytic in a strip in the neighborhood SY of the real line.The parameterζ 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 being a complex domain, one studies solutions satisfying the following condition:(4.2)ψ(x+1,ζ)=ψ(x,ζ+ε),∀ζ∈D.
The aim of the WKB method is to construct solutions to (4.1) satisfying (4.2) and that have simple asymptotic behavior when ɛ 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 will use these solutions to compute the monodromy matrix; we consider V and W as fixed and construct the WKB objects and solutions in a uniform way for energies near E.


### 4.1. Standard Behavior of Consistent Solutions

We start by defining another analytic object central to the complex WKB method, thecanonical Bloch solutions. Then, we describe the standard behavior of the solutions.


#### 4.1.1. Canonical Bloch Solutions

To describe the asymptotic formulae of the complex WKB method, one needs to construct Bloch solutions to the following equation(4.3)-d2dx2ψ(x)+V(x)ψ(x)=ℰ(ζ)ψ(x),ℰ(ζ)=E-W(ζ),x∈ℝ,
that are moreover analytic in ζ on a given regular domain.Letζ0 be a regular point (i.e., ζ0 is not a branch point of κ). Let ℰ0=ℰ(ζ0). Assume that ℰ0∉P∪Q. Let U0 be a sufficiently small neighborhood of ℰ0, and let V0 be a neighborhood of ζ0 such that ℰ(V0)⊂U0. In U0, 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±, and the corresponding branches of Ω (see Section 2.4.). For ζ∈V0, we set(4.4)Ψ±(x,ζ)=q(ℰ)e∫ℰ0ℰΩ±ψ±(x,ℰ),q(ℰ)=k′(ℰ),ℰ=ℰ(ζ).
The functions Ψ± are called the canonical Bloch solutions normalized at the pointζ0.The properties of the differentialΩ imply that the solutions Ψ± can be analytically continued from V0 to any regular domain containing V0.The Wronskian ofΨ± satisfies (see [13])(4.5)w(Ψ+(·,ζ),Ψ-(·,ζ))=w(Ψ+(·,ζ0),Ψ-(·,ζ0))=k′(ℰ0)w(ψ+(·,ζ0),ψ-(·,ζ0)).
For ℰ0∉Q∪{El}, the Wronskian w(Ψ+(·,ζ),Ψ-(·,ζ)) is nonzero.


### 4.2. Solutions Having Standard Asymptotic Behavior

FixE=E0. Let D be a regular domain (i.e., D⊂D(W), and simply connected set containing no branch points of κ). Fix ζ0∈D so that ℰ(ζ0)∉P∪Q. Let κ be a continuous branch of the complex momentum in D, and let Ψ± be the canonical Bloch solutions normalized at ζ0 defined on D and indexed so that κ is the Quasimomentum for Ψ+.Definition 4.1.
Letσ∈{+,-}. We say that, in D, a consistent solution f has standard behavior (or standard asymptotics) if  (i)
there existsV0, a complex neighborhood of E0, and X>0 such that f is defined and satisfies (4.1) and (4.2) for any (x,ζ,E)∈(-X,X)×D×V0;(ii)
f is analytic in ζ∈D and in E∈V0;(iii)
for any compact set K⊂D, there exists V⊂V0, a neighborhood of E0, such that, for (x,ζ,E)∈(-X,X)×K×V, f has the uniform asymptotic:
(4.6)f=eσ(i/ε)∫ζ0ζκ(u)du(Ψσ+o(1))asεtendsto0;(iv)
this asymptotic can be differentiated once inx without loosing its uniformity properties. We set
(4.7)f*(x,ζ,E,ε)=f(x,ζ¯,E¯,ε)¯.We callζ0 the normalization point for f. To say that a consistent solution f has standard behavior, we will use the following notation: (4.8)f~exp(σiε∫ζ0ζκ(u)du)Ψσ.


### 4.3. Some Results on the Continuation of Asymptotics

#### 4.3.1. Description of the Stokes Lines near[φ1-(E),φ1+(E)]

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, 15]. The integral ζ↦∫ζκ(u)du has the same branch points as the complex momentum. Let ζ0 be one of them. Consider the curves beginning at ζ0 and described by the following equation:(4.9)Im∫ζ0ζ(κ(ξ)-κ(ζ0))dξ=0.
These curves are the Stokes lines beginning at ζ0. According to (2.8), the Stokes line definition is independent of the choice of the branch of κ.Assume thatW′(ζ0)≠0. Equation (2.9) implies that there are exactly three Stokes lines beginning at ζ0. The angle between any two of them at this point is equal to 2π/3. Indeed for ζ near ζ0, we have (4.10)κ(ζ)=κ(ζ0)+cζ-ζ0(1+o(1)).
So (4.11)∫ζ0ζ(κ(ζ)-κ(ζ0))dζ=c(ζ-ζ0)3/2(1+o(1)).


#### 4.3.3. Stokes Lines forE0∈J

We describe the Stokes lines beginning atφ1-(E) and φ1+(E). Since W is real on ℝ, the set of the Stokes lines is symmetric with respect to the real line.First,κ1 is real on the interval [φ1-(E),φ1+(E)]; therefore this set is a Stokes line starting at φ1-(E). The two other Stokes lines beginning at φ1-(E) are symmetric with respect to the real axis. We denote by σ1-(E) the Stokes line going downward and by σ1¯(E) its symmetric. Similarly, we denote by σ1+(E) and σ1+¯(E) the two other Stokes lines starting at φ1+(E), and σ1+(E) goes upward. These Stokes lines are represented in Figure 5.Figure 5
Stokes lines forE∈J.Lemma 4.2.
The Stokes linesσ1-(E) and σ1+(E) satisfy the following properties.  (i)
The Stokes linesσ1-(E) and σ1+(E) stay vertical.(ii)
σ1-(E) and σ1+(E) do not intersect one another.The proof of this lemma is similar to the studies done in [5, 14, 17, 18]. We do not give the details.


### 4.4. Construction of a Consistent Basis near[φ1-(E),φ1+(E)]

We recall this result, proved in [17].Proposition 4.3 (see [17]).
FixE0∈J, and let κ1 be a continuous determination of the complex momentum on [φ1-(E),φ1+(E)]. There exists a real number Y>0, a complex neighborhood 𝒱1 of E0, and a consistent basis (f1,(f1)*) of solutions of (4.1) such that f1 has the standard asymptotic behavior:
(4.12)f1~e(i/ε)∫κ̃1Ψ1+,(4.13)f1*(x,ζ,E)=f(x,ζ¯,E¯)¯
to the left of σ1-(E)∪σ1+(E¯)¯ (resp., to the right of σ1+(E)∪σ1-(E¯)¯). The determination κ̃1 is the continuation of κ1 through {ζ∈(σ1-(E)∪σ1+(E¯)¯)∩SY;Imκ1(ζ)>0}(resp., through {ζ∈(σ1-(E)∪σ1+(E¯)¯)∩SY;Imκ1(ζ)<0}).We mimic the analysis done in [5, Section  5]. Precisely, we start by a local construction of the solution f using canonical domain; then, we apply continuation tools, that is, the rectangle lemma, the adjacent domain principle, and the Stokes Lemma.


## 4.1. Standard Behavior of Consistent Solutions

We start by defining another analytic object central to the complex WKB method, thecanonical Bloch solutions. Then, we describe the standard behavior of the solutions.


### 4.1.1. Canonical Bloch Solutions

To describe the asymptotic formulae of the complex WKB method, one needs to construct Bloch solutions to the following equation(4.3)-d2dx2ψ(x)+V(x)ψ(x)=ℰ(ζ)ψ(x),ℰ(ζ)=E-W(ζ),x∈ℝ,
that are moreover analytic in ζ on a given regular domain.Letζ0 be a regular point (i.e., ζ0 is not a branch point of κ). Let ℰ0=ℰ(ζ0). Assume that ℰ0∉P∪Q. Let U0 be a sufficiently small neighborhood of ℰ0, and let V0 be a neighborhood of ζ0 such that ℰ(V0)⊂U0. In U0, 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±, and the corresponding branches of Ω (see Section 2.4.). For ζ∈V0, we set(4.4)Ψ±(x,ζ)=q(ℰ)e∫ℰ0ℰΩ±ψ±(x,ℰ),q(ℰ)=k′(ℰ),ℰ=ℰ(ζ).
The functions Ψ± are called the canonical Bloch solutions normalized at the pointζ0.The properties of the differentialΩ imply that the solutions Ψ± can be analytically continued from V0 to any regular domain containing V0.The Wronskian ofΨ± satisfies (see [13])(4.5)w(Ψ+(·,ζ),Ψ-(·,ζ))=w(Ψ+(·,ζ0),Ψ-(·,ζ0))=k′(ℰ0)w(ψ+(·,ζ0),ψ-(·,ζ0)).
For ℰ0∉Q∪{El}, the Wronskian w(Ψ+(·,ζ),Ψ-(·,ζ)) is nonzero.


## 4.1.1. Canonical Bloch Solutions

To describe the asymptotic formulae of the complex WKB method, one needs to construct Bloch solutions to the following equation(4.3)-d2dx2ψ(x)+V(x)ψ(x)=ℰ(ζ)ψ(x),ℰ(ζ)=E-W(ζ),x∈ℝ,
that are moreover analytic in ζ on a given regular domain.Letζ0 be a regular point (i.e., ζ0 is not a branch point of κ). Let ℰ0=ℰ(ζ0). Assume that ℰ0∉P∪Q. Let U0 be a sufficiently small neighborhood of ℰ0, and let V0 be a neighborhood of ζ0 such that ℰ(V0)⊂U0. In U0, 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±, and the corresponding branches of Ω (see Section 2.4.). For ζ∈V0, we set(4.4)Ψ±(x,ζ)=q(ℰ)e∫ℰ0ℰΩ±ψ±(x,ℰ),q(ℰ)=k′(ℰ),ℰ=ℰ(ζ).
The functions Ψ± are called the canonical Bloch solutions normalized at the pointζ0.The properties of the differentialΩ imply that the solutions Ψ± can be analytically continued from V0 to any regular domain containing V0.The Wronskian ofΨ± satisfies (see [13])(4.5)w(Ψ+(·,ζ),Ψ-(·,ζ))=w(Ψ+(·,ζ0),Ψ-(·,ζ0))=k′(ℰ0)w(ψ+(·,ζ0),ψ-(·,ζ0)).
For ℰ0∉Q∪{El}, the Wronskian w(Ψ+(·,ζ),Ψ-(·,ζ)) is nonzero.


## 4.2. Solutions Having Standard Asymptotic Behavior

FixE=E0. Let D be a regular domain (i.e., D⊂D(W), and simply connected set containing no branch points of κ). Fix ζ0∈D so that ℰ(ζ0)∉P∪Q. Let κ be a continuous branch of the complex momentum in D, and let Ψ± be the canonical Bloch solutions normalized at ζ0 defined on D and indexed so that κ is the Quasimomentum for Ψ+.Definition 4.1.
Letσ∈{+,-}. We say that, in D, a consistent solution f has standard behavior (or standard asymptotics) if  (i)
there existsV0, a complex neighborhood of E0, and X>0 such that f is defined and satisfies (4.1) and (4.2) for any (x,ζ,E)∈(-X,X)×D×V0;(ii)
f is analytic in ζ∈D and in E∈V0;(iii)
for any compact set K⊂D, there exists V⊂V0, a neighborhood of E0, such that, for (x,ζ,E)∈(-X,X)×K×V, f has the uniform asymptotic:
(4.6)f=eσ(i/ε)∫ζ0ζκ(u)du(Ψσ+o(1))asεtendsto0;(iv)
this asymptotic can be differentiated once inx without loosing its uniformity properties. We set
(4.7)f*(x,ζ,E,ε)=f(x,ζ¯,E¯,ε)¯.We callζ0 the normalization point for f. To say that a consistent solution f has standard behavior, we will use the following notation: (4.8)f~exp(σiε∫ζ0ζκ(u)du)Ψσ.


## 4.3. Some Results on the Continuation of Asymptotics

### 4.3.1. Description of the Stokes Lines near[φ1-(E),φ1+(E)]

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, 15]. The integral ζ↦∫ζκ(u)du has the same branch points as the complex momentum. Let ζ0 be one of them. Consider the curves beginning at ζ0 and described by the following equation:(4.9)Im∫ζ0ζ(κ(ξ)-κ(ζ0))dξ=0.
These curves are the Stokes lines beginning at ζ0. According to (2.8), the Stokes line definition is independent of the choice of the branch of κ.Assume thatW′(ζ0)≠0. Equation (2.9) implies that there are exactly three Stokes lines beginning at ζ0. The angle between any two of them at this point is equal to 2π/3. Indeed for ζ near ζ0, we have (4.10)κ(ζ)=κ(ζ0)+cζ-ζ0(1+o(1)).
So (4.11)∫ζ0ζ(κ(ζ)-κ(ζ0))dζ=c(ζ-ζ0)3/2(1+o(1)).


### 4.3.3. Stokes Lines forE0∈J

We describe the Stokes lines beginning atφ1-(E) and φ1+(E). Since W is real on ℝ, the set of the Stokes lines is symmetric with respect to the real line.First,κ1 is real on the interval [φ1-(E),φ1+(E)]; therefore this set is a Stokes line starting at φ1-(E). The two other Stokes lines beginning at φ1-(E) are symmetric with respect to the real axis. We denote by σ1-(E) the Stokes line going downward and by σ1¯(E) its symmetric. Similarly, we denote by σ1+(E) and σ1+¯(E) the two other Stokes lines starting at φ1+(E), and σ1+(E) goes upward. These Stokes lines are represented in Figure 5.Figure 5
Stokes lines forE∈J.Lemma 4.2.
The Stokes linesσ1-(E) and σ1+(E) satisfy the following properties.  (i)
The Stokes linesσ1-(E) and σ1+(E) stay vertical.(ii)
σ1-(E) and σ1+(E) do not intersect one another.The proof of this lemma is similar to the studies done in [5, 14, 17, 18]. We do not give the details.


## 4.3.1. Description of the Stokes Lines near[φ1-(E),φ1+(E)]

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, 15]. The integral ζ↦∫ζκ(u)du has the same branch points as the complex momentum. Let ζ0 be one of them. Consider the curves beginning at ζ0 and described by the following equation:(4.9)Im∫ζ0ζ(κ(ξ)-κ(ζ0))dξ=0.
These curves are the Stokes lines beginning at ζ0. According to (2.8), the Stokes line definition is independent of the choice of the branch of κ.Assume thatW′(ζ0)≠0. Equation (2.9) implies that there are exactly three Stokes lines beginning at ζ0. The angle between any two of them at this point is equal to 2π/3. Indeed for ζ near ζ0, we have (4.10)κ(ζ)=κ(ζ0)+cζ-ζ0(1+o(1)).
So (4.11)∫ζ0ζ(κ(ζ)-κ(ζ0))dζ=c(ζ-ζ0)3/2(1+o(1)).


## 4.3.3. Stokes Lines forE0∈J

We describe the Stokes lines beginning atφ1-(E) and φ1+(E). Since W is real on ℝ, the set of the Stokes lines is symmetric with respect to the real line.First,κ1 is real on the interval [φ1-(E),φ1+(E)]; therefore this set is a Stokes line starting at φ1-(E). The two other Stokes lines beginning at φ1-(E) are symmetric with respect to the real axis. We denote by σ1-(E) the Stokes line going downward and by σ1¯(E) its symmetric. Similarly, we denote by σ1+(E) and σ1+¯(E) the two other Stokes lines starting at φ1+(E), and σ1+(E) goes upward. These Stokes lines are represented in Figure 5.Figure 5
Stokes lines forE∈J.Lemma 4.2.
The Stokes linesσ1-(E) and σ1+(E) satisfy the following properties.  (i)
The Stokes linesσ1-(E) and σ1+(E) stay vertical.(ii)
σ1-(E) and σ1+(E) do not intersect one another.The proof of this lemma is similar to the studies done in [5, 14, 17, 18]. We do not give the details.


## 4.4. Construction of a Consistent Basis near[φ1-(E),φ1+(E)]

We recall this result, proved in [17].Proposition 4.3 (see [17]).
FixE0∈J, and let κ1 be a continuous determination of the complex momentum on [φ1-(E),φ1+(E)]. There exists a real number Y>0, a complex neighborhood 𝒱1 of E0, and a consistent basis (f1,(f1)*) of solutions of (4.1) such that f1 has the standard asymptotic behavior:
(4.12)f1~e(i/ε)∫κ̃1Ψ1+,(4.13)f1*(x,ζ,E)=f(x,ζ¯,E¯)¯
to the left of σ1-(E)∪σ1+(E¯)¯ (resp., to the right of σ1+(E)∪σ1-(E¯)¯). The determination κ̃1 is the continuation of κ1 through {ζ∈(σ1-(E)∪σ1+(E¯)¯)∩SY;Imκ1(ζ)>0}(resp., through {ζ∈(σ1-(E)∪σ1+(E¯)¯)∩SY;Imκ1(ζ)<0}).We mimic the analysis done in [5, Section  5]. Precisely, we start by a local construction of the solution f using canonical domain; then, we apply continuation tools, that is, the rectangle lemma, the adjacent domain principle, and the Stokes Lemma.


## 5. The Proof of Theorem3.4

The Proof of Theorem3.4 follows the same ideas as the computations given in  [5, Section  10.2]. Below we give the details for the proof of (3.13), (3.14), and (3.15).


### 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 solutionf of (3.2) satisfying (3.3).To use the complex WKB method, we perform the following change of variable in (3.2):(5.1)x-z→x,εz→ζ.
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.Consider the basis(f1,f1*) constructed around [φ1-,φ1+] as in Proposition 4.3. Then the monodromy matrix associated to the basis (f1,f1*) (defined in Section 3.2.1) satisfies
(5.2)(f1(x,ζ,E,ε)f1*(x,ζ,E,ε))=M(ζ,E,ε)(f1(x,ζ-2π,E,ε)f1*(x,ζ-2π,E,ε)).
The aim of this section is the computation of M(ζ,E,ε). The definition of the monodromy matrix implies that(5.3)A(ζ)=M1,1=w(f1(x+2π,ζ)f1*(x,ζ))w(f1(x,ζ),f1*(x,ζ)),B(ζ)=M12=w(f1(x,ζ),f1(x,ζ+2π))w(f1(x,ζ),f1*(x,ζ)).
This gives that the monodromy matrix is analytic in ζ in the strip SY and in E in a constant neighborhood of E0. By the definition of f1 we get that A and B are ε-periodic in ζ∈SY. This is an immediate consequence of the properties of f1.Therefore, we will compute the Fourier series ofA and B. The strategy of the computation is based on the ideas of [5] and we first recall some notions presented there. We refer the reader to this paper for more details.Leth and g have a standard asymptotic behavior in regular domains Dh and Dg and solutions of (4.1): (5.4)h~e(i/ε)∫ζhζκhdζψh(x,ζ),g~e(i/ε)∫ζgζκgdζψg(x,ζ).
Here, κh (resp., κg) is an analytic branch of the complex momentum in Dh (resp., Dg), Ψh (resp., Ψg) is the canonical Bloch solution defined on Dh (resp., Dg), and ζh (resp., ζg) is the normalization point for h (resp., g).As the solutionsh and g satisfy the consistency condition (4.2), their Wronskian is ε-periodic in ζ.


#### 5.1.1. Arcs

We assume thatDg∩Dh contains a simply connected domain D̃. Let γ be a regular curve going from ζg to ζh in the following way: staying in Dg, it goes from ζg to some point in D̃, and then, staying in Dh, it goes to ζh. We say that γ is an arc associated to the triple h,g, and D̃.AsD̃ is simply connected, all the arcs associated to the triple, h,g, and D̃. As D̃ 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,D̃).We continueκh and κg analytically along γ(h,g,D̃). From the properties of κ, we deduce that, for V a small neighborhood of γ, one has(5.5)κg(ζ)=σκh+2πm,forsomem∈ℤ,σ∈{1,-1}.σ=σ,h,g,D̃ is called the signature of γ(h,g,D̃), and m=m(h,g,D̃) the index of γ(h,g,D̃).


#### 5.1.2. The Meeting Domain

LetD̃ be as above. We call D̃ ameeting domain, if, in D̃, the functions Imκh and Imκg do not vanish and are of opposite signs.Note that, for small values ofε, whether ζ↦g(x,ζ) and ζ↦h(x,ζ) increase or decrease is determined by the exponential factor e∫ζgζκgdζ and e∫ζhζκhdζ. So, roughly, in a meeting domain, along the lines Imζ=Const, the solutions h and g increase in opposite directions.


#### 5.1.3. The Action and the Amplitude of an Arc

We call the integral(5.6)S(h,g,D̃)=∫γ(h,g,D̃)κgdζ,
the action of the arc γ=γ(h,g,D̃). Clearly, the action takes the same value for equivalent arcs.Assume thatℰ(ζ)∉P∪Q along γ(h,f,D̃). Consider the function qg=k′(ℰ) and the 1-form Ωg(ℰ(ζ)) in the definition of Ψg. Continue them analytically along γ. We set
(5.7)A(h,g,γ)=(qgqh)|ζ=ζhe∫γωg.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,D̃) coincide.


### 5.2. Results on the Fourier Coefficients

We recall the following result from [5].Proposition 5.1.
Letd=d(h,g) be a meeting domain for h and g, and let m=m(h,g,d) be the corresponding index. Then
(5.8)w(h,g)=wme(2πim/ε)(ζ-ζh)(1+o(1));ζ∈S(d),
where wm is the constant given by
(5.9)wm=A(h,g,d)e(i/ε)S(h,g,d)w(ψ+(·,ζh),ψ-(·,ζh)).
Here ψ+=ψh and ψ- is complementary to ψ+. The asymptotic (5.8) is uniform in ζ and E when ζ stays in a fixed compact of S(d) and E in a small enough neighborhood of E0.


#### 5.2.1. The Indexm

Letζ0 be a regular point. Consider a regular curve γ going from ζ0 to ζ0+2π. Let κ be a branch of the complex momentum that is continuous on γ. We call the couple (γ,κ) a period. Let (γ1,κ1) and (γ2,κ2) be two periods. Assume that one can continuously deform γ1 into γ2 without intersecting any branching point. By this we define an analytic continuation of κ1 to γ2. If the analytic continuation coincides with κ2, we say that the periods are equivalent.Consider the branchκ along the curve γ of a period (γ,κ). In a neighborhood of ζ0, the starting point γ, one has (5.10)κ(ζ+2π)=σκ(ζ)+2πm,σ∈{∓1},m∈ℤ.
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.Recall that(5.11)𝒢=∪k∈ℤ{∪j=1N𝒢j+2πk}
is the preimage with respect to ℰ of the union of the spectral gaps of H0. One has the following.Lemma 5.2 (see [10]).
Let(γ,κ) be a period such that γ starts at a point ζ0∉𝒢.Assume that γ intersects 𝒢 exactly n times (n∈ℕ) and that at all intersection points, W′≠0. Let r1,…,rn be the values that Reκ takes consecutively at these intersection points as ζ moves along γ from ζ0 to ζ0+2π. Then,
(5.12)σ(γ,κ)=(-1)n,m(γ,κ)=1π(rn-rn-1+rn-2-⋯+(-1)n-1r1)=(-1)n-1π(r1-r2+⋯+(-1)n-1rn).


### 5.3. The Fourier Coefficients

#### 5.3.1. ForB

By (5.3), we have to compute w(f(·,ζ),f(·,ζ+2π)). With this aim in view, we apply the construction done in Section 5.1 with (5.13)h(x,ζ)=f(x,ζ),g(x,ζ)=(Tf)(x,ζ),with(Tf)(x,ζ)=f(x,ζ+2π),(5.14)Dh=𝒟,Dg=𝒟-2π,ζh=ζ0,ζg=ζ0-2π,(5.15)∀ζ∈Dh,κh(ζ)=κ(ζ),∀ζ∈Dg,κg(ζ)=κ(ζ+2π).
We will start by the following.


#### 5.3.2. Above the Real Line

We take the meeting domainD0 as the subdomain of the strip 0<Imζ<Y between the Stokes lines σ1+-2π and σ1+. In this domain we have Imκg=-Imκh<0. Indeed we notice that the sign of Imκ changes to opposite one as ζ intersects 𝒵 at a point where W′≠0 taking into account that κg(ζ)=κh(ζ+2π), and to go from ζ to ζ+2π one has to intersect ℬ1.The arcγ0 connects the point ζg to ζh. By (5.14), this defines the period (γ0,κg). Using (5.15), one gets that m(f(·,ζ),f(·,ζ+2π,D0))=m(γ0+2π,κ) [5].We use Lemma5.2 to compute the index. To do this, we have to compute Reκ at the intersection of γ0+2π and 𝒢.Asζ→Reκ(ζ) is constant on any connected component 𝒢j of 𝒢. Let us start by defining the index λj+ of 𝒢j, the result of the alternated character of the coefficients (…,+,-,+,-,…) due to the crossing of 𝒢j (see Lemma 5.2). We notice that λj+∈{-1,0,1}.We set(5.16)Nj=1+∑i=1j-1(∑l=1ni(ol-1)).
Here nj is the number of extremum in 𝒢j, and ol is the order of the lth extremum. The following relations hold:(5.17)λ1+=1+(-1)∑l=1n1(ol-1)2,
and for 2≤j≤N,(5.18)λj+=(-1)Nj(-1+(-1)∑l=1nj(ol-1)2).
Without loss of generality we assume that Reκ=0 on (0,φ1-). By the above notation, we get that(5.19)m(γ0,κ)=(-1)n(p1λ1++(p1+p2)λ2++⋯+(∑i=1jpi)λj++⋯+(∑i=1Npi)λN+)=P+.
Here pi is the index of ℬi and n=NN.Now using (5.3) for B and Proposition 5.1 we get(5.20)B=A(f,T(f),D̃0)e(i/ε)S(f,T(f),D̃0)·e-2iπP+ζ0/ε=B+(E,ε)·e-(i/2πε)P+ζ0,T(f)(x,ζ)=f(x,ζ+2π).


#### 5.3.3. Below the Real Line

Below the real line, we take the domain of the strip-Y<Imζ<0 located between the stokes line σ1--2π and σ1- as a regular domain, which we denote by D̃0. We set γ̃0=γ(f,T(f),D̃1); then it defines a period, and so m(f,T(f),D̃1)=m(γ̃0+2π,κ). The curve defining a period equivalent to (γ̃0+2π,κ) is represented in Figure 6.Figure 6
Periods equivalent toγ0 and γ1.The computation of the indexλj- of 𝒢j gives in this case that(5.21)λ1-=-1+(-1)∑i=1n1(ol-1)2,(5.22)λj-=(-1)Nj(-1+(-1)∑l=1l=nj(ol-1)2).
By this notation we get that(5.23)m(γ0,κ)=(-1)n(p1λ1-+(p1+p2)λ2-+⋯+(∑i=1jpi)λj-+⋯+(∑i=1Npi)λN-)=P-.
Using (5.3) for B and Proposition 5.1 we get(5.24)BP-=A(f,T(f),D̃1)e(i/ε)(S(f,Tf,D̃1)-2πP-ζ0)=B-(E,ε)·e(i/ε)2πP-ζ0,(Tf)(x,ζ)=f(x,ζ+2π).


### 5.4. ForA

For the computation ofA using (5.3), we have to compute w(f*(·,ζ),f(·,ζ+2π)). It suffices to apply the method presented in Section 5.1 with (5.25)h(x,ζ)=f*(x,ζ),g(x,ζ)=(Tf)(x,ζ),with(Tf)(x,ζ)=f(x,ζ+2π),(5.26)Dh=𝒟*,Dg=𝒟-2π,ζh=ζ0,ζg=ζ0-2π,∀ζ∈Dh,κh(ζ)=-κ¯(ζ¯),∀ζ∈Dg,κg(ζ)=κ(ζ+2π).


#### 5.4.1. Above the Real Line

In this case,D̃0, the meeting domain, is the subdomain of the strip {0<Imζ<Y} located between the lines σ1+-2π and σ1-¯, the symmetric to σ1- with respect to ℝ. The arc γ(f*,T(f),D̃0) defines a period (γ̃0,κg); in Figure 7 we represent the curve γ̃0+2π.Figure 7
Periods equivalent toγ̃0 and γ̃1.Similarly to the computation ofB, we define the index βj+ of 𝒢j, and in this case, we have
(5.27)β1+=-1+(-1)∑i=1n1(ol-1)2,(5.28)βj+=(-1)Nj(-1+(-1)∑l=1l=nj(ol-1)2).
By this notation we get that(5.29)m(γ0,κ)=(-1)n(p1β1++(p1+p2)β2++⋯+(∑i=1jpi)βj++⋯+(∑i=1Npi)βN+)=Q+.
Using (5.3) for A and Proposition 5.1 we get that (5.30)m(f*,Tf,D̃0)=m(γ̃0,κ)=Q+,(5.31)A=A(f*,Tf,D̃0)e(i/ε)(S(f*,Tf,D̃0)-2πQ+ζ0)=A+(E,ε)·e-(i/ε)2πQ+,(Tf)(x,ζ)=f(x,ζ+2π).


#### 5.4.2. Below the Real Line

In this case,D̃1, the meeting domain, is the subdomain of the band {-Y<Imζ<0} located between the lines σ1+¯, symmetric of σ1+ with respect to ℝ, and σ1--2π. The arc γ(h,g,D̃1) defines a period (γ̃1,κg); the curve γ̃1+2π is represented in Figure 7. One obtains that(5.32)β1-=1+(-1)∑l=1n1(ol-1)2,
and for 2≤j≤N,(5.33)βj-=(-1)Nj(-1+(-1)∑l=1nj(ol-1)2).
So by the above notation we get that(5.34)m(γ0,κ)=(-1)n(p1β1-+(p1+p2)β2-+⋯+(∑i=1j‍pi)βj-+⋯+(∑i=1Npi)βN-)=Q-.
One obtains that m(f*,T(f),D̃1)=m(γ̃1+2π,κ)=Q-, and(5.35)A=A(f*,Tf,D̃1)e(i/ε)(S(f*,Tf,D̃1)-2πQ-ζ0)=A-(E,ε)·e(i/ε)2πQ-ζ0,(Tf)(x,ζ)=f(x,ζ+2π).


#### 5.4.3. The Proof of (3.15)

Letg∈𝒢. Let V(g) be a complex neighborhood of g sufficiently small so that it contains only two branch points of κ, namely, the ends of g. Let γg be a curve in V(g)∖g; the tunneling action is defined as S(g)=i∮γgκdζ. From [5], we recall the following lemma.Lemma 5.3 (see [5]).
LetE∈J. If γg is positively oriented, then
(5.36)S(g)=±2∫g±i0Imκdζ.
Here, in the left-hand side, one integrates in the increasing direction on the real axis.Below we give details only forA+; the same way could be used to obtain the result for A- and B-±.We recall that the definition of the amplitude of an arcγ is given in (5.7) and the coefficient A+ is defined in (5.31). This definition implies that A(f*,T(f),D̃0) is independent of ε, continuous in E, and does not vanish; so we get that for V0 a small constant neighborhood of E0∈J,∃C1,C2>0 such that (5.37)C1<|A(f*,T(f),D̃0)|<C2.
Let γ=γ(f*,T(f),D̃0) be the arc going near the real line and going around the branch points between ζ0 and ζ0+2π (the beginning and the end of γ) along infinitesimally small circles. One has(5.38)|exp(iεS(f*,T(f),D̃0))|=|exp(-1ε∫γImκgdζ)|=exp(-1ε∑g∈𝒢∫gImκgdζ).Imκg<0, inside each of the intervals of integration. Using (5.36), the expression -2∫gImκdζ is equal to S(g), the tunneling action, because S(g) is taken positive. Then we get (5.39)|exp(iεS((f*,T(f),D̃0)))|=∏g∈𝒢(t(g))-1.
This ends the proof of (3.15) for A+.


## 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 solutionf of (3.2) satisfying (3.3).To use the complex WKB method, we perform the following change of variable in (3.2):(5.1)x-z→x,εz→ζ.
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.Consider the basis(f1,f1*) constructed around [φ1-,φ1+] as in Proposition 4.3. Then the monodromy matrix associated to the basis (f1,f1*) (defined in Section 3.2.1) satisfies
(5.2)(f1(x,ζ,E,ε)f1*(x,ζ,E,ε))=M(ζ,E,ε)(f1(x,ζ-2π,E,ε)f1*(x,ζ-2π,E,ε)).
The aim of this section is the computation of M(ζ,E,ε). The definition of the monodromy matrix implies that(5.3)A(ζ)=M1,1=w(f1(x+2π,ζ)f1*(x,ζ))w(f1(x,ζ),f1*(x,ζ)),B(ζ)=M12=w(f1(x,ζ),f1(x,ζ+2π))w(f1(x,ζ),f1*(x,ζ)).
This gives that the monodromy matrix is analytic in ζ in the strip SY and in E in a constant neighborhood of E0. By the definition of f1 we get that A and B are ε-periodic in ζ∈SY. This is an immediate consequence of the properties of f1.Therefore, we will compute the Fourier series ofA and B. The strategy of the computation is based on the ideas of [5] and we first recall some notions presented there. We refer the reader to this paper for more details.Leth and g have a standard asymptotic behavior in regular domains Dh and Dg and solutions of (4.1): (5.4)h~e(i/ε)∫ζhζκhdζψh(x,ζ),g~e(i/ε)∫ζgζκgdζψg(x,ζ).
Here, κh (resp., κg) is an analytic branch of the complex momentum in Dh (resp., Dg), Ψh (resp., Ψg) is the canonical Bloch solution defined on Dh (resp., Dg), and ζh (resp., ζg) is the normalization point for h (resp., g).As the solutionsh and g satisfy the consistency condition (4.2), their Wronskian is ε-periodic in ζ.


### 5.1.1. Arcs

We assume thatDg∩Dh contains a simply connected domain D̃. Let γ be a regular curve going from ζg to ζh in the following way: staying in Dg, it goes from ζg to some point in D̃, and then, staying in Dh, it goes to ζh. We say that γ is an arc associated to the triple h,g, and D̃.AsD̃ is simply connected, all the arcs associated to the triple, h,g, and D̃. As D̃ 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,D̃).We continueκh and κg analytically along γ(h,g,D̃). From the properties of κ, we deduce that, for V a small neighborhood of γ, one has(5.5)κg(ζ)=σκh+2πm,forsomem∈ℤ,σ∈{1,-1}.σ=σ,h,g,D̃ is called the signature of γ(h,g,D̃), and m=m(h,g,D̃) the index of γ(h,g,D̃).


### 5.1.2. The Meeting Domain

LetD̃ be as above. We call D̃ ameeting domain, if, in D̃, the functions Imκh and Imκg do not vanish and are of opposite signs.Note that, for small values ofε, whether ζ↦g(x,ζ) and ζ↦h(x,ζ) increase or decrease is determined by the exponential factor e∫ζgζκgdζ and e∫ζhζκhdζ. So, roughly, in a meeting domain, along the lines Imζ=Const, the solutions h and g increase in opposite directions.


### 5.1.3. The Action and the Amplitude of an Arc

We call the integral(5.6)S(h,g,D̃)=∫γ(h,g,D̃)κgdζ,
the action of the arc γ=γ(h,g,D̃). Clearly, the action takes the same value for equivalent arcs.Assume thatℰ(ζ)∉P∪Q along γ(h,f,D̃). Consider the function qg=k′(ℰ) and the 1-form Ωg(ℰ(ζ)) in the definition of Ψg. Continue them analytically along γ. We set
(5.7)A(h,g,γ)=(qgqh)|ζ=ζhe∫γωg.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,D̃) coincide.


## 5.1.1. Arcs

We assume thatDg∩Dh contains a simply connected domain D̃. Let γ be a regular curve going from ζg to ζh in the following way: staying in Dg, it goes from ζg to some point in D̃, and then, staying in Dh, it goes to ζh. We say that γ is an arc associated to the triple h,g, and D̃.AsD̃ is simply connected, all the arcs associated to the triple, h,g, and D̃. As D̃ 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,D̃).We continueκh and κg analytically along γ(h,g,D̃). From the properties of κ, we deduce that, for V a small neighborhood of γ, one has(5.5)κg(ζ)=σκh+2πm,forsomem∈ℤ,σ∈{1,-1}.σ=σ,h,g,D̃ is called the signature of γ(h,g,D̃), and m=m(h,g,D̃) the index of γ(h,g,D̃).


## 5.1.2. The Meeting Domain

LetD̃ be as above. We call D̃ ameeting domain, if, in D̃, the functions Imκh and Imκg do not vanish and are of opposite signs.Note that, for small values ofε, whether ζ↦g(x,ζ) and ζ↦h(x,ζ) increase or decrease is determined by the exponential factor e∫ζgζκgdζ and e∫ζhζκhdζ. So, roughly, in a meeting domain, along the lines Imζ=Const, the solutions h and g increase in opposite directions.


## 5.1.3. The Action and the Amplitude of an Arc

We call the integral(5.6)S(h,g,D̃)=∫γ(h,g,D̃)κgdζ,
the action of the arc γ=γ(h,g,D̃). Clearly, the action takes the same value for equivalent arcs.Assume thatℰ(ζ)∉P∪Q along γ(h,f,D̃). Consider the function qg=k′(ℰ) and the 1-form Ωg(ℰ(ζ)) in the definition of Ψg. Continue them analytically along γ. We set
(5.7)A(h,g,γ)=(qgqh)|ζ=ζhe∫γωg.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,D̃) coincide.


## 5.2. Results on the Fourier Coefficients

We recall the following result from [5].Proposition 5.1.
Letd=d(h,g) be a meeting domain for h and g, and let m=m(h,g,d) be the corresponding index. Then
(5.8)w(h,g)=wme(2πim/ε)(ζ-ζh)(1+o(1));ζ∈S(d),
where wm is the constant given by
(5.9)wm=A(h,g,d)e(i/ε)S(h,g,d)w(ψ+(·,ζh),ψ-(·,ζh)).
Here ψ+=ψh and ψ- is complementary to ψ+. The asymptotic (5.8) is uniform in ζ and E when ζ stays in a fixed compact of S(d) and E in a small enough neighborhood of E0.


### 5.2.1. The Indexm

Letζ0 be a regular point. Consider a regular curve γ going from ζ0 to ζ0+2π. Let κ be a branch of the complex momentum that is continuous on γ. We call the couple (γ,κ) a period. Let (γ1,κ1) and (γ2,κ2) be two periods. Assume that one can continuously deform γ1 into γ2 without intersecting any branching point. By this we define an analytic continuation of κ1 to γ2. If the analytic continuation coincides with κ2, we say that the periods are equivalent.Consider the branchκ along the curve γ of a period (γ,κ). In a neighborhood of ζ0, the starting point γ, one has (5.10)κ(ζ+2π)=σκ(ζ)+2πm,σ∈{∓1},m∈ℤ.
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.Recall that(5.11)𝒢=∪k∈ℤ{∪j=1N𝒢j+2πk}
is the preimage with respect to ℰ of the union of the spectral gaps of H0. One has the following.Lemma 5.2 (see [10]).
Let(γ,κ) be a period such that γ starts at a point ζ0∉𝒢.Assume that γ intersects 𝒢 exactly n times (n∈ℕ) and that at all intersection points, W′≠0. Let r1,…,rn be the values that Reκ takes consecutively at these intersection points as ζ moves along γ from ζ0 to ζ0+2π. Then,
(5.12)σ(γ,κ)=(-1)n,m(γ,κ)=1π(rn-rn-1+rn-2-⋯+(-1)n-1r1)=(-1)n-1π(r1-r2+⋯+(-1)n-1rn).


## 5.2.1. The Indexm

Letζ0 be a regular point. Consider a regular curve γ going from ζ0 to ζ0+2π. Let κ be a branch of the complex momentum that is continuous on γ. We call the couple (γ,κ) a period. Let (γ1,κ1) and (γ2,κ2) be two periods. Assume that one can continuously deform γ1 into γ2 without intersecting any branching point. By this we define an analytic continuation of κ1 to γ2. If the analytic continuation coincides with κ2, we say that the periods are equivalent.Consider the branchκ along the curve γ of a period (γ,κ). In a neighborhood of ζ0, the starting point γ, one has (5.10)κ(ζ+2π)=σκ(ζ)+2πm,σ∈{∓1},m∈ℤ.
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.Recall that(5.11)𝒢=∪k∈ℤ{∪j=1N𝒢j+2πk}
is the preimage with respect to ℰ of the union of the spectral gaps of H0. One has the following.Lemma 5.2 (see [10]).
Let(γ,κ) be a period such that γ starts at a point ζ0∉𝒢.Assume that γ intersects 𝒢 exactly n times (n∈ℕ) and that at all intersection points, W′≠0. Let r1,…,rn be the values that Reκ takes consecutively at these intersection points as ζ moves along γ from ζ0 to ζ0+2π. Then,
(5.12)σ(γ,κ)=(-1)n,m(γ,κ)=1π(rn-rn-1+rn-2-⋯+(-1)n-1r1)=(-1)n-1π(r1-r2+⋯+(-1)n-1rn).


## 5.3. The Fourier Coefficients

### 5.3.1. ForB

By (5.3), we have to compute w(f(·,ζ),f(·,ζ+2π)). With this aim in view, we apply the construction done in Section 5.1 with (5.13)h(x,ζ)=f(x,ζ),g(x,ζ)=(Tf)(x,ζ),with(Tf)(x,ζ)=f(x,ζ+2π),(5.14)Dh=𝒟,Dg=𝒟-2π,ζh=ζ0,ζg=ζ0-2π,(5.15)∀ζ∈Dh,κh(ζ)=κ(ζ),∀ζ∈Dg,κg(ζ)=κ(ζ+2π).
We will start by the following.


### 5.3.2. Above the Real Line

We take the meeting domainD0 as the subdomain of the strip 0<Imζ<Y between the Stokes lines σ1+-2π and σ1+. In this domain we have Imκg=-Imκh<0. Indeed we notice that the sign of Imκ changes to opposite one as ζ intersects 𝒵 at a point where W′≠0 taking into account that κg(ζ)=κh(ζ+2π), and to go from ζ to ζ+2π one has to intersect ℬ1.The arcγ0 connects the point ζg to ζh. By (5.14), this defines the period (γ0,κg). Using (5.15), one gets that m(f(·,ζ),f(·,ζ+2π,D0))=m(γ0+2π,κ) [5].We use Lemma5.2 to compute the index. To do this, we have to compute Reκ at the intersection of γ0+2π and 𝒢.Asζ→Reκ(ζ) is constant on any connected component 𝒢j of 𝒢. Let us start by defining the index λj+ of 𝒢j, the result of the alternated character of the coefficients (…,+,-,+,-,…) due to the crossing of 𝒢j (see Lemma 5.2). We notice that λj+∈{-1,0,1}.We set(5.16)Nj=1+∑i=1j-1(∑l=1ni(ol-1)).
Here nj is the number of extremum in 𝒢j, and ol is the order of the lth extremum. The following relations hold:(5.17)λ1+=1+(-1)∑l=1n1(ol-1)2,
and for 2≤j≤N,(5.18)λj+=(-1)Nj(-1+(-1)∑l=1nj(ol-1)2).
Without loss of generality we assume that Reκ=0 on (0,φ1-). By the above notation, we get that(5.19)m(γ0,κ)=(-1)n(p1λ1++(p1+p2)λ2++⋯+(∑i=1jpi)λj++⋯+(∑i=1Npi)λN+)=P+.
Here pi is the index of ℬi and n=NN.Now using (5.3) for B and Proposition 5.1 we get(5.20)B=A(f,T(f),D̃0)e(i/ε)S(f,T(f),D̃0)·e-2iπP+ζ0/ε=B+(E,ε)·e-(i/2πε)P+ζ0,T(f)(x,ζ)=f(x,ζ+2π).


### 5.3.3. Below the Real Line

Below the real line, we take the domain of the strip-Y<Imζ<0 located between the stokes line σ1--2π and σ1- as a regular domain, which we denote by D̃0. We set γ̃0=γ(f,T(f),D̃1); then it defines a period, and so m(f,T(f),D̃1)=m(γ̃0+2π,κ). The curve defining a period equivalent to (γ̃0+2π,κ) is represented in Figure 6.Figure 6
Periods equivalent toγ0 and γ1.The computation of the indexλj- of 𝒢j gives in this case that(5.21)λ1-=-1+(-1)∑i=1n1(ol-1)2,(5.22)λj-=(-1)Nj(-1+(-1)∑l=1l=nj(ol-1)2).
By this notation we get that(5.23)m(γ0,κ)=(-1)n(p1λ1-+(p1+p2)λ2-+⋯+(∑i=1jpi)λj-+⋯+(∑i=1Npi)λN-)=P-.
Using (5.3) for B and Proposition 5.1 we get(5.24)BP-=A(f,T(f),D̃1)e(i/ε)(S(f,Tf,D̃1)-2πP-ζ0)=B-(E,ε)·e(i/ε)2πP-ζ0,(Tf)(x,ζ)=f(x,ζ+2π).


## 5.3.1. ForB

By (5.3), we have to compute w(f(·,ζ),f(·,ζ+2π)). With this aim in view, we apply the construction done in Section 5.1 with (5.13)h(x,ζ)=f(x,ζ),g(x,ζ)=(Tf)(x,ζ),with(Tf)(x,ζ)=f(x,ζ+2π),(5.14)Dh=𝒟,Dg=𝒟-2π,ζh=ζ0,ζg=ζ0-2π,(5.15)∀ζ∈Dh,κh(ζ)=κ(ζ),∀ζ∈Dg,κg(ζ)=κ(ζ+2π).
We will start by the following.


## 5.3.2. Above the Real Line

We take the meeting domainD0 as the subdomain of the strip 0<Imζ<Y between the Stokes lines σ1+-2π and σ1+. In this domain we have Imκg=-Imκh<0. Indeed we notice that the sign of Imκ changes to opposite one as ζ intersects 𝒵 at a point where W′≠0 taking into account that κg(ζ)=κh(ζ+2π), and to go from ζ to ζ+2π one has to intersect ℬ1.The arcγ0 connects the point ζg to ζh. By (5.14), this defines the period (γ0,κg). Using (5.15), one gets that m(f(·,ζ),f(·,ζ+2π,D0))=m(γ0+2π,κ) [5].We use Lemma5.2 to compute the index. To do this, we have to compute Reκ at the intersection of γ0+2π and 𝒢.Asζ→Reκ(ζ) is constant on any connected component 𝒢j of 𝒢. Let us start by defining the index λj+ of 𝒢j, the result of the alternated character of the coefficients (…,+,-,+,-,…) due to the crossing of 𝒢j (see Lemma 5.2). We notice that λj+∈{-1,0,1}.We set(5.16)Nj=1+∑i=1j-1(∑l=1ni(ol-1)).
Here nj is the number of extremum in 𝒢j, and ol is the order of the lth extremum. The following relations hold:(5.17)λ1+=1+(-1)∑l=1n1(ol-1)2,
and for 2≤j≤N,(5.18)λj+=(-1)Nj(-1+(-1)∑l=1nj(ol-1)2).
Without loss of generality we assume that Reκ=0 on (0,φ1-). By the above notation, we get that(5.19)m(γ0,κ)=(-1)n(p1λ1++(p1+p2)λ2++⋯+(∑i=1jpi)λj++⋯+(∑i=1Npi)λN+)=P+.
Here pi is the index of ℬi and n=NN.Now using (5.3) for B and Proposition 5.1 we get(5.20)B=A(f,T(f),D̃0)e(i/ε)S(f,T(f),D̃0)·e-2iπP+ζ0/ε=B+(E,ε)·e-(i/2πε)P+ζ0,T(f)(x,ζ)=f(x,ζ+2π).


## 5.3.3. Below the Real Line

Below the real line, we take the domain of the strip-Y<Imζ<0 located between the stokes line σ1--2π and σ1- as a regular domain, which we denote by D̃0. We set γ̃0=γ(f,T(f),D̃1); then it defines a period, and so m(f,T(f),D̃1)=m(γ̃0+2π,κ). The curve defining a period equivalent to (γ̃0+2π,κ) is represented in Figure 6.Figure 6
Periods equivalent toγ0 and γ1.The computation of the indexλj- of 𝒢j gives in this case that(5.21)λ1-=-1+(-1)∑i=1n1(ol-1)2,(5.22)λj-=(-1)Nj(-1+(-1)∑l=1l=nj(ol-1)2).
By this notation we get that(5.23)m(γ0,κ)=(-1)n(p1λ1-+(p1+p2)λ2-+⋯+(∑i=1jpi)λj-+⋯+(∑i=1Npi)λN-)=P-.
Using (5.3) for B and Proposition 5.1 we get(5.24)BP-=A(f,T(f),D̃1)e(i/ε)(S(f,Tf,D̃1)-2πP-ζ0)=B-(E,ε)·e(i/ε)2πP-ζ0,(Tf)(x,ζ)=f(x,ζ+2π).


## 5.4. ForA

For the computation ofA using (5.3), we have to compute w(f*(·,ζ),f(·,ζ+2π)). It suffices to apply the method presented in Section 5.1 with (5.25)h(x,ζ)=f*(x,ζ),g(x,ζ)=(Tf)(x,ζ),with(Tf)(x,ζ)=f(x,ζ+2π),(5.26)Dh=𝒟*,Dg=𝒟-2π,ζh=ζ0,ζg=ζ0-2π,∀ζ∈Dh,κh(ζ)=-κ¯(ζ¯),∀ζ∈Dg,κg(ζ)=κ(ζ+2π).


### 5.4.1. Above the Real Line

In this case,D̃0, the meeting domain, is the subdomain of the strip {0<Imζ<Y} located between the lines σ1+-2π and σ1-¯, the symmetric to σ1- with respect to ℝ. The arc γ(f*,T(f),D̃0) defines a period (γ̃0,κg); in Figure 7 we represent the curve γ̃0+2π.Figure 7
Periods equivalent toγ̃0 and γ̃1.Similarly to the computation ofB, we define the index βj+ of 𝒢j, and in this case, we have
(5.27)β1+=-1+(-1)∑i=1n1(ol-1)2,(5.28)βj+=(-1)Nj(-1+(-1)∑l=1l=nj(ol-1)2).
By this notation we get that(5.29)m(γ0,κ)=(-1)n(p1β1++(p1+p2)β2++⋯+(∑i=1jpi)βj++⋯+(∑i=1Npi)βN+)=Q+.
Using (5.3) for A and Proposition 5.1 we get that (5.30)m(f*,Tf,D̃0)=m(γ̃0,κ)=Q+,(5.31)A=A(f*,Tf,D̃0)e(i/ε)(S(f*,Tf,D̃0)-2πQ+ζ0)=A+(E,ε)·e-(i/ε)2πQ+,(Tf)(x,ζ)=f(x,ζ+2π).


### 5.4.2. Below the Real Line

In this case,D̃1, the meeting domain, is the subdomain of the band {-Y<Imζ<0} located between the lines σ1+¯, symmetric of σ1+ with respect to ℝ, and σ1--2π. The arc γ(h,g,D̃1) defines a period (γ̃1,κg); the curve γ̃1+2π is represented in Figure 7. One obtains that(5.32)β1-=1+(-1)∑l=1n1(ol-1)2,
and for 2≤j≤N,(5.33)βj-=(-1)Nj(-1+(-1)∑l=1nj(ol-1)2).
So by the above notation we get that(5.34)m(γ0,κ)=(-1)n(p1β1-+(p1+p2)β2-+⋯+(∑i=1j‍pi)βj-+⋯+(∑i=1Npi)βN-)=Q-.
One obtains that m(f*,T(f),D̃1)=m(γ̃1+2π,κ)=Q-, and(5.35)A=A(f*,Tf,D̃1)e(i/ε)(S(f*,Tf,D̃1)-2πQ-ζ0)=A-(E,ε)·e(i/ε)2πQ-ζ0,(Tf)(x,ζ)=f(x,ζ+2π).


### 5.4.3. The Proof of (3.15)

Letg∈𝒢. Let V(g) be a complex neighborhood of g sufficiently small so that it contains only two branch points of κ, namely, the ends of g. Let γg be a curve in V(g)∖g; the tunneling action is defined as S(g)=i∮γgκdζ. From [5], we recall the following lemma.Lemma 5.3 (see [5]).
LetE∈J. If γg is positively oriented, then
(5.36)S(g)=±2∫g±i0Imκdζ.
Here, in the left-hand side, one integrates in the increasing direction on the real axis.Below we give details only forA+; the same way could be used to obtain the result for A- and B-±.We recall that the definition of the amplitude of an arcγ is given in (5.7) and the coefficient A+ is defined in (5.31). This definition implies that A(f*,T(f),D̃0) is independent of ε, continuous in E, and does not vanish; so we get that for V0 a small constant neighborhood of E0∈J,∃C1,C2>0 such that (5.37)C1<|A(f*,T(f),D̃0)|<C2.
Let γ=γ(f*,T(f),D̃0) be the arc going near the real line and going around the branch points between ζ0 and ζ0+2π (the beginning and the end of γ) along infinitesimally small circles. One has(5.38)|exp(iεS(f*,T(f),D̃0))|=|exp(-1ε∫γImκgdζ)|=exp(-1ε∑g∈𝒢∫gImκgdζ).Imκg<0, inside each of the intervals of integration. Using (5.36), the expression -2∫gImκdζ is equal to S(g), the tunneling action, because S(g) is taken positive. Then we get (5.39)|exp(iεS((f*,T(f),D̃0)))|=∏g∈𝒢(t(g))-1.
This ends the proof of (3.15) for A+.


## 5.4.1. Above the Real Line

In this case,D̃0, the meeting domain, is the subdomain of the strip {0<Imζ<Y} located between the lines σ1+-2π and σ1-¯, the symmetric to σ1- with respect to ℝ. The arc γ(f*,T(f),D̃0) defines a period (γ̃0,κg); in Figure 7 we represent the curve γ̃0+2π.Figure 7
Periods equivalent toγ̃0 and γ̃1.Similarly to the computation ofB, we define the index βj+ of 𝒢j, and in this case, we have
(5.27)β1+=-1+(-1)∑i=1n1(ol-1)2,(5.28)βj+=(-1)Nj(-1+(-1)∑l=1l=nj(ol-1)2).
By this notation we get that(5.29)m(γ0,κ)=(-1)n(p1β1++(p1+p2)β2++⋯+(∑i=1jpi)βj++⋯+(∑i=1Npi)βN+)=Q+.
Using (5.3) for A and Proposition 5.1 we get that (5.30)m(f*,Tf,D̃0)=m(γ̃0,κ)=Q+,(5.31)A=A(f*,Tf,D̃0)e(i/ε)(S(f*,Tf,D̃0)-2πQ+ζ0)=A+(E,ε)·e-(i/ε)2πQ+,(Tf)(x,ζ)=f(x,ζ+2π).


## 5.4.2. Below the Real Line

In this case,D̃1, the meeting domain, is the subdomain of the band {-Y<Imζ<0} located between the lines σ1+¯, symmetric of σ1+ with respect to ℝ, and σ1--2π. The arc γ(h,g,D̃1) defines a period (γ̃1,κg); the curve γ̃1+2π is represented in Figure 7. One obtains that(5.32)β1-=1+(-1)∑l=1n1(ol-1)2,
and for 2≤j≤N,(5.33)βj-=(-1)Nj(-1+(-1)∑l=1nj(ol-1)2).
So by the above notation we get that(5.34)m(γ0,κ)=(-1)n(p1β1-+(p1+p2)β2-+⋯+(∑i=1j‍pi)βj-+⋯+(∑i=1Npi)βN-)=Q-.
One obtains that m(f*,T(f),D̃1)=m(γ̃1+2π,κ)=Q-, and(5.35)A=A(f*,Tf,D̃1)e(i/ε)(S(f*,Tf,D̃1)-2πQ-ζ0)=A-(E,ε)·e(i/ε)2πQ-ζ0,(Tf)(x,ζ)=f(x,ζ+2π).


## 5.4.3. The Proof of (3.15)

Letg∈𝒢. Let V(g) be a complex neighborhood of g sufficiently small so that it contains only two branch points of κ, namely, the ends of g. Let γg be a curve in V(g)∖g; the tunneling action is defined as S(g)=i∮γgκdζ. From [5], we recall the following lemma.Lemma 5.3 (see [5]).
LetE∈J. If γg is positively oriented, then
(5.36)S(g)=±2∫g±i0Imκdζ.
Here, in the left-hand side, one integrates in the increasing direction on the real axis.Below we give details only forA+; the same way could be used to obtain the result for A- and B-±.We recall that the definition of the amplitude of an arcγ is given in (5.7) and the coefficient A+ is defined in (5.31). This definition implies that A(f*,T(f),D̃0) is independent of ε, continuous in E, and does not vanish; so we get that for V0 a small constant neighborhood of E0∈J,∃C1,C2>0 such that (5.37)C1<|A(f*,T(f),D̃0)|<C2.
Let γ=γ(f*,T(f),D̃0) be the arc going near the real line and going around the branch points between ζ0 and ζ0+2π (the beginning and the end of γ) along infinitesimally small circles. One has(5.38)|exp(iεS(f*,T(f),D̃0))|=|exp(-1ε∫γImκgdζ)|=exp(-1ε∑g∈𝒢∫gImκgdζ).Imκg<0, inside each of the intervals of integration. Using (5.36), the expression -2∫gImκdζ is equal to S(g), the tunneling action, because S(g) is taken positive. Then we get (5.39)|exp(iεS((f*,T(f),D̃0)))|=∏g∈𝒢(t(g))-1.
This ends the proof of (3.15) for A+.



---

*Source: 145436-2010-05-25.xml*