On the Singular Spectrum for Adiabatic Quasiperiodic Schrödinger Operators

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).

1. Introduction

We consider the following Schrödinger equation: (𝐻_𝑧,𝜀𝑑𝜓)(𝑥)=−²𝑑𝑥²𝜓(𝑥)+[𝑉(𝑥−𝑧)+𝑊(𝜀𝑥)]𝜓(𝑥)=𝐸𝜓(𝑥),𝑥∈ℝ, (1.1) 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 𝐻_𝑧,𝜀 can be regarded as an adiabatic perturbation of the periodic operator 𝐻₀:𝐻₀𝑑=−²𝑑𝑥²+𝑉(𝑥). (1.2) 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
Let ℰ(𝜅) be the dispersion relation associated to 𝐻₀. Consider the complex and real-isoenergy curves  Γ_ℂ and Γ_ℝ defined by Γ_ℂ(𝐸)={(𝜅,𝜁)∈ℂ²Γ;ℰ(𝜅)+𝑊(𝜁)=𝐸};_ℝ(𝐸)={(𝜅,𝜁)∈ℝ²;ℰ(𝜅)+𝑊(𝜁)=𝐸}. (1.3) Notice that the iso-energy curves Γ_ℂ(𝐸) and Γ_ℝ(𝐸) are 2𝜋-periodic in 𝜁 and 𝜅 and Γ_ℂ is the Riemann surface uniformizing 𝜅.

The real iso-energy curve has a well-known role for adiabatic problems [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 interval 𝐸−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 Γ_ℂ; 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 interval 𝐸−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 Γ_ℝ; the vertical loops are situated in Γ_ℂ. 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.

When 𝑊 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 Γ_ℝ, 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: 𝑉(𝑥+1)=𝑉(𝑥),𝑊(𝑥+2𝜋)=𝑊(𝑥),∀𝑥∈ℝ. (1.4)(H2)𝑉 is real-valued and locally square-integrable.(H3)𝑊 is real analytic in the strip 𝑆_𝑌={𝑧∈ℂ;|Im𝑧|<𝑌}.

We define𝑊₋=inf_𝑡∈ℝ𝑊(𝑡),𝑊₊=sup_𝑡∈ℝ𝑊(𝑡). (1.5) We define the spectral window 𝒲(𝐸) by 𝒲(𝐸)=𝐸−𝑊(ℝ)=[𝐸−𝑊₊,𝐸−𝑊₋]. (1.6)

1.1.2. Assumptions on the Energy Region

To describe the energy regions where we study the spectral properties, we consider the periodic Schrödinger operator 𝐻₀ 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 [𝐸_2𝑛+1,𝐸_2𝑛+2] for 𝑛∈ℕ, such that 𝐸₁<𝐸₂≤𝐸₃<𝐸₄,…,𝐸_2𝑛≤𝐸_2𝑛+1<𝐸_2𝑛+2,… and 𝐸_𝑛→+∞,𝑛→+∞. The points (𝐸_𝑗)_𝑗∈ℕ are the eigenvalues of the self-adjoint operator obtained by considering 𝐻₀ defined by (1.2) and acting in 𝐿²([0,2]) with periodic boundary conditions (see [6, 7]). The intervals [𝐸_2𝑛+1,𝐸_2𝑛+2],𝑛∈ℕ, are the spectral bands, and the intervals (𝐸_2𝑛,𝐸_2𝑛+1),𝑛∈ℕ^∗, the spectral gaps. When 𝐸_2𝑛<𝐸_2𝑛+1, one says that the 𝑛th gap is open; when [𝐸_2𝑛−1,𝐸_2𝑛] 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 [𝐸_{2(𝑛+𝑗)−1},𝐸_2(𝑛+𝑗)],𝑗=1,2,…,𝑁, are isolated. (H4b)For all 𝐸∈𝐽, these bands are contained in the interior of 𝒲(𝐸). (H4c):For all 𝐸∈𝐽, the rest of the spectrum of 𝐻₀ is outside of 𝒲(𝐸).

Remark 1.1. The geometric assumption assures that the iso-energy curve Γ_ℝ(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. Let 𝐽 be a real compact interval. We assume that (H1)-(H4) are satisfied. For 𝜀>0 sufficiently small, for almost all 𝑧∈ℝ, one has 𝜎_𝑎𝑐(𝐻_𝑧,𝜀)∩𝐽=∅. (1.7) Here 𝜎_𝑎𝑐(𝐻_𝑧,𝜀) is the absolutely continuous spectrum of the family of (𝐻_𝑧,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(𝐻_𝑧,𝜀)⊂Σ, and for any 𝐾⊂Σ, compact, there exists 𝐶>0 such that for all 𝜀 sufficiently small and all 𝐸∈𝐾, 𝜎(𝐻_𝑧,𝜀)∩(𝐸−𝐶𝜀^1/2,𝐸+𝐶𝜀^1/2)≠∅. (1.8) So, for an interval 𝐽 as in Theorem 1.2, we have 𝜎(𝐻_𝑧,𝜀)∩𝐽≠∅. (1.9)
(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, 𝐿²_loc- 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 Set 𝑊⁻¹(ℝ)

As 𝐸∈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 of 𝑊 of order 𝑛_𝑖 on the real line, say 𝜁_𝑖. Near 𝜁_𝑖, the set 𝑊⁻¹(ℝ) consists of a real segment and of 𝑛_𝑖−1 complex curves symmetric with respect to the real axis and intersecting the real axis only on 𝜁_𝑖. The angle between two neighboring curves is equal to 𝜋/𝑛_𝑖. Let 𝑌>0. We set 𝑆_𝑌={−𝑌≤Im𝜁≤𝑌}. We assume that 𝑌 is so small that (i)𝑆_𝑌 is contained in the domain of analyticity of 𝑊;(ii)the set 𝑊⁻¹(ℝ)∩𝑆_𝑌 consists of the real line and of the complex lines passing through the real extrema of 𝑊.

An example of subset 𝑊⁻¹(ℝ) is shown in Figure 3.

Figure 3 

A subset of 𝑊−1(ℝ)∩𝑆𝑌.

Figure 4 

Bands, gaps, and the action of mapping of the quasimomentum 𝑘𝑝 on some curves.

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

For all 𝐸∈𝐽, we write (𝐸−𝑊)⁻¹(𝜎(𝐻₀))∩ℝ=^𝑁_{𝑘∈ℤ𝑗=1}{[𝜑⁻_𝑗(𝐸),𝜑⁺_𝑗(𝐸)]+2𝑘𝜋}, (2.1) with the following properties. (i)𝜑⁻₁(𝐸)<𝜑⁺₁(𝐸)<𝜑⁻₂(𝐸)<⋯<𝜑⁻_𝑁(𝐸)<𝜑⁺_𝑁(𝐸),0<𝜑⁺_𝑁(𝐸)−𝜑⁻₁(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 [𝜑⁻₁(𝐸),𝜑⁺₁(𝐸)] is associated to a connected component of Γ_ℝ(𝐸) which is an unbounded vertical curve. See Remark 2.3. (iii)We generally define the following 𝜑⁻_{𝑗+𝑘+𝑁}(𝐸)=𝜑⁻_𝑗(𝐸)+2𝑘𝜋,∀𝑗∈{1,…,𝑁},∀𝑘∈ℤ. (2.2) We set ℬ_𝑗(𝐸) = [𝜑⁻_𝑗(𝐸),𝜑⁺_𝑗(𝐸)],𝒵 = ∪^𝑁_𝑗=1ℬ_𝑗(𝐸),𝒢_𝑗(𝐸) = ]𝜑⁺_𝑗(𝐸),𝜑⁻_𝑗+1(𝐸)[, and 𝒢_𝑁(𝐸) =  (𝜑⁺_𝑁(𝐸),𝜑⁻₁(𝐸)+2𝜋) = (𝜑⁺_𝑁(𝐸),2𝜋)∪(2𝜋,2𝜋+𝜑⁻₁(𝐸)). Let (𝜁^𝑗_𝑖)_{1≤𝑖≤𝑝𝑗} be the extrema of 𝑊 in 𝒢_𝑗. We recall that 𝑛^𝑗_𝑖 is the order of 𝜁^𝑗_𝑖.

We have the following description.

Lemma 2.1. Fix [𝐴,𝐵] a compact interval of ℝ.
There exists a finite number 𝑝 of real extrema of 𝑊 in [𝐴,𝐵]. (i)If 𝑝=0, there exists 𝑌>0 such that (𝐸₀−𝑊)⁻¹(ℝ)∩{𝜁∈𝑆_𝑌;Re𝜁∈[𝐴,𝐵]}=[𝐴,𝐵]. (2.3)
(ii)For 𝑝>0, one denotes by {𝜁₁,…,𝜁_𝑝} the real extrema of 𝑊 in [𝐴,𝐵]. There exists 𝑌>0 and a sequence {Σ¹_𝑖,…,Σ^𝑛𝑖_𝑖⁻¹}_{𝑖∈{1…𝑝}} of disjoint and strictly vertical lines of ℂ₊ starting at 𝜁_𝑖 such that (𝐸₀−𝑊)⁻¹(ℝ)∩{𝜁∈𝑆_𝑌;Re𝜁∈[𝐴,𝐵]}=[𝐴,𝐵]^𝑝_𝑖=1(^𝑛𝑖−1_𝑘=1(Σ^𝑘_𝑖∪Σ^𝑘_𝑖)). (2.4)

2.2. Bloch Solutions

Let 𝜓 be a solution of the equation−𝑑²𝑑𝑥²𝜓(𝑥,ℰ)+𝑉(𝑥)𝜓(𝑥,ℰ)=ℰ𝜓(𝑥,ℰ),𝑥∈ℝ (2.5) satisfying the relation𝜓(𝑥+1,ℰ)=𝜆(ℰ)𝜓(𝑥,ℰ) (2.6) 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 Section 1.1.2, we denote the spectral bands of the periodic Schrödinger operator by [𝐸_2𝑛+1,𝐸_2𝑛+2],𝑛∈ℕ. Consider 𝒮_± two copies of the complex plane ℰ∈ℂ cut along the spectral bands. Paste them together to get a Riemann surface with square root branch points. We denote this Riemann surface by 𝒮.

One can construct a Bloch solution 𝜓(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.

For ℰ∈𝒮, we denote by ℰ the point on 𝒮 different from ℰ and having the same projection on ℂ as ℰ. We let 𝜓(𝑥,ℰ)=𝜓(𝑥,ℰ),ℰ∈𝒮. (2.7) The function 𝜓(𝑥,ℰ) is another Bloch solution of (2.5). Except at the edges of the spectrum, the functions 𝜓 and 𝜓 are linearly independent solutions of (2.5). In the spectral gaps, 𝜓 and 𝜓 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 solution 𝜓(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.

Let 𝐷 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: 𝑘(ℰ)=±𝑘₀(ℰ)+2𝜋𝑙,𝑙∈ℤ. (2.8) 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 𝑘_𝑝.

The function 𝑘_𝑝 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 [𝐸_2𝑛−1,𝐸_2𝑛], the 𝑛th spectral band, is mapped on the interval [𝜋(𝑛−1),𝜋𝑛]. Along any open gap, Re𝑘_𝑝(ℰ+𝑖0) is constant, and Im𝑘_𝑝(ℰ+𝑖0) is positive and has only one nondegenerate maximum.

Consider ℂ₀, 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↦𝑘_𝑝(ℰ).

All the branch points of 𝑘_𝑝 are of square root type. Let 𝐸_𝑙 be a branch point of 𝑘_𝑝. In a sufficiently small neighborhood of 𝐸_𝑙, the function 𝑘_𝑝 is analytic in 𝐸−𝐸_𝑙, and𝑘_𝑝(ℰ)−𝑘_𝑝(𝐸_𝑙)=𝑐_𝑙ℰ−𝐸_𝑙+𝑜(ℰ−𝐸_𝑙),𝑐_𝑙≠0. (2.9) Finally, we note that the main branch can be continued analytically to the complex plane cut along (−∞,𝐸₁] and the spectral gaps ]𝐸_2𝑛,𝐸_2𝑛+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 differential Ω=𝜔𝑑ℰ is meromorphic; its poles are the points of 𝑃∪𝑄, where 𝑃 is the set of poles of ℰ↦𝜓(𝑥,ℰ), and 𝑄 is the set of zeros of 𝑘^′;(ii)all the poles of Ω are simple; (iii)if the residue of Ω at a point 𝑝 is denoted by res_𝑝Ω, one has res_𝑝Ω=1,∀𝑝∈𝑃⧵𝑄,res_𝑞1Ω=−2,∀𝑞∈𝑄⧵𝑃,res_𝑟1Ω=2,∀𝑟∈𝑃∩𝑄. (2.10)(iv) if ℰ∈𝒮 projects into a gap, then 𝜔(ℰ)∈ℝ.(v)if ℰ∈𝒮 projects inside a band, then 𝜔(ℰ)=𝜔(ℰ).

2.4.1. The Complex Momentum

It is the main analytic object of the complex WKB method. Let 𝜁∈𝑆_𝑌. We define 𝜅, in 𝒟(𝑊) the domain of analyticity of 𝑊, by𝜅(𝜁)=𝑘(𝐸−𝑊(𝜁)). (2.11) 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 relations𝐸−𝑊(𝜁)=𝐸_𝑙,𝑙=1,2,3,…. (2.12) Let 𝜁₀ be a branch point of 𝜅. If 𝑊^′(𝜁₀)≠0, then 𝜁₀ is a branch point of square root type.

If 𝐷⊂𝒟(𝑊) is a simply connected set containing no branch points of 𝜅, we call it regular. Let 𝜅_𝑝 be a branch of the complex momentum analytic in a regular domain 𝐷. All the other branches that are analytic in 𝐷 are described by the following formulae:𝜅^±_𝑚=±𝜅_𝑝+2𝜋𝑚. (2.13) Here ± and 𝑚∈ℤ are indexing the branches.

2.4.2. Index of an Interval [𝜑⁻_𝑗(𝐸),𝜑⁺_𝑗(𝐸)]

Fix 𝑗∈{1,…,𝑁}. Fix a continuous branch 𝜅_𝑗 of the complex momentum on [𝜑⁻_𝑗(𝐸),𝜑⁺_𝑗(𝐸)]. We define𝜅⁺_𝑗=𝜅_𝑗(𝜑⁺_𝑗);𝜅⁻_𝑗=𝜅_𝑗(𝜑⁻_𝑗);𝑝_𝑗𝜋=𝜅⁺_𝑗−𝜅⁻_𝑗. (2.14)𝑝_𝑗 is the index of [𝜑⁻_𝑗(𝐸),𝜑⁺_𝑗(𝐸)] associated to 𝜅_𝑗.

Let us give some properties of 𝑝_𝑗.

Lemma 2.2. Assume that (H4) is satisfied. The indices 𝑝_𝑗 have the following properties. (1)For 𝑗∈{1,…,𝑁},𝑝_𝑗∈{−1,0,1}.(2)∑^𝑁_𝑗=1|𝑝_𝑗|∈2ℕ.

Proof. The points (𝐸−𝑊)(𝜑⁻_𝑗(𝐸)) and (𝐸−𝑊)(𝜑⁺_𝑗(𝐸)) are the ends of a band of 𝜎(𝐻₀): they are distinct or coincide. If they coincide, that is, if (𝐸−𝑊)(𝜑⁻_𝑗(𝐸))=(𝐸−𝑊)(𝜑⁺_𝑗(𝐸)), the index 𝑝_𝑗 satisfies 𝑝_𝑗=0. Else, we consider 𝑘_𝑗 the branch of the Quasimomentum associated to 𝜅_𝑗. We have that |𝑘_𝑗((𝐸−𝑊)(𝜑⁻_𝑗(𝐸)))−𝑘_𝑗((𝐸−𝑊)(𝜑⁺_𝑗(𝐸)))|=𝜋, (2.15) see (2.8), and |𝑝_𝑗|=1.
Let us prove point (2).As for any 𝑗∈{1,…,𝑁}, we have |𝑝_𝑗|=𝑝_𝑗mod2. We write ^𝑁_𝑗=1|𝑝_𝑗|≡^𝑁_𝑗=1𝑝_𝑗mod2,_𝑁𝑗=1𝑝_𝑗=^𝑁_𝑗=1𝜅_𝑗(𝜑⁺_𝑗)−𝜅_𝑗(𝜑⁻_𝑗)𝜋=𝜅_𝑁(𝜑⁺_𝑁)−𝜅₁(𝜑⁻₁)𝜋+^𝑁−1_𝑗=1𝜅_𝑗+1(𝜑⁻_𝑗+1)−𝜅_𝑗(𝜑⁺_𝑗)𝜋. (2.16) For 𝑗∈{1,…,𝑁−1},(𝐸−𝑊)(𝜑⁻_𝑗+1(𝐸)) and (𝐸−𝑊)(𝜑⁺_𝑗(𝐸)) are the ends of a same gap and 𝑘_𝑗+1((𝐸−𝑊)(𝜑⁻_𝑗+1(𝐸)))≡𝑘_𝑗((𝐸−𝑊)(𝜑⁺_𝑗(𝐸)))[2𝜋]. (2.17) By periodicity, (𝐸−𝑊)(𝜑⁻₁(𝐸)) and (𝐸−𝑊)(𝜑⁺_𝑁(𝐸)) are the ends of the same gap.
This ends the proof of Lemma 2.2.

If 𝑝_𝑗≠0, we say that we cross a band. In this case, (𝐸−𝑊)(𝜑⁻_𝑗)≠(𝐸−𝑊)(𝜑⁺_𝑗) and the associated connected component of the iso-energy curve is unbounded vertically.

We notice that 𝑝₁≠0, and thus ∑^𝑁_𝑗=1|𝑝_𝑗|>0.

Remark 2.3. We can choose the determination of 𝜅 such that 𝑝₁=1.

2.4.3. Tunneling Coefficients

For 𝑗∈{1,…,𝑁}, we denote by 𝛾_𝑗 a smooth closed curve that goes once around [𝜑⁺_𝑗(𝐸),𝜑⁻_𝑗+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 𝑆_𝑗 given by𝑆_𝑗(𝐸)=𝑖_𝛾𝑗𝜅(𝜁)𝑑𝜁,∀𝑗∈{1,…𝑁}. (2.18) It is straightforward to prove that for 𝐸∈𝐽, each of these actions is real and nonzero and that 𝑆_𝑗(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 𝑡_𝑗=𝑒^{−(1/2𝜀)𝑆𝑗(𝐸)}. (2.19)𝑆_𝑗(𝐸) is called the tunneling action; we choose the branch 𝜅 so that on 𝐽,𝑆_𝑗(𝐸) is positive. We define 𝑇(𝐸,𝜀)=^𝑁_𝑗=1𝑡_𝑗. (2.20) So we get 𝑇(𝐸,𝜀)=𝑒^{−∑𝑁𝑗=1𝑆𝑗(𝐸)/2𝜀}. (2.21)𝑆_𝑗(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 Theorem 1.2

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 𝜀Θ(𝐸,𝜀)=2𝜋^𝑁_𝑗=11ln𝑡_𝑗1+𝑜(1)=4𝜋^𝑁_𝑗=1𝑆_𝑗(𝐸)+𝑜(1). (3.1)

This theorem implies that if 𝜀/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

Fix 𝐸∈ℝ. Consider the family of differential equations indexed by 𝑧∈ℝ:𝑑(−²𝑑𝑥²+𝑉(𝑥−𝑧)+𝑊(𝜀𝑥))𝜓(𝑥)=𝐸𝜓(𝑥). (3.2)

Definition 3.2. We say that (𝜓_𝑖)_𝑖∈{1,2} is a consistent basis of solutions to (3.2) if the two functions ((𝑥,𝑧)↦𝜓_𝑖(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 ∀𝑥∈ℝ,∀𝑧∈ℝ,∀𝑖∈{1,2},𝜓_𝑖(𝑥,𝑧+1,𝐸)=𝜓_𝑖(𝑥,𝑧,𝐸). (3.3)

We refer the reader to [5, 10] about the existence and details on consistent basis of solutions to (3.2).

The functions ((𝑥,𝑧)↦𝜓_𝑖(𝑥+2𝜋/𝜀,𝑧+2𝜋/𝜀,𝐸))_𝑖∈{1,2} being also solutions of (3.2), we get the relationΨ(𝑥+2𝜋𝜀,𝑧+2𝜋𝜀,𝐸)=𝑀(𝑧,𝐸)Ψ(𝑥,𝑧,𝐸), (3.4) where (i)Ψ(𝑥,𝑧,𝐸)=(^𝜓1_𝜓^{(𝑥,𝑧,𝐸)}_{2(𝑥,𝑧,𝐸)}),(ii)𝑀(𝑧,𝐸) is a 2×2-matrix with coefficients independent of 𝑥.

The matrix 𝑀 is called the monodromy matrix associated to the consistent basis (𝜓_1,2). We recall the following properties of this matrix:det𝑀(𝑧,𝐸)≡1,𝑀(𝑧+1,𝐸)=𝑀(𝑧,𝐸),∀𝑧∈ℝ. (3.5) 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↦𝑀, and ℎ>0 irrational. Consider the finite difference equation:𝐹_𝑛+1=𝑀(𝑧+𝑛ℎ)𝐹_𝑛∀𝑛∈ℤ,𝐹_𝑛∈ℂ². (3.6) 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) is𝜃(𝑀,ℎ)=lim_𝑁↦+∞1𝑁log‖𝑃_𝑁(𝑧,ℎ)‖, (3.7) where the matrix cocycle (𝑃_𝑁(𝑧,ℎ))_𝑁∈ℕ is defined as𝑃_𝑁(𝑧,ℎ)=𝑀(𝑧+𝑁ℎ)⋅𝑀(𝑧+(𝑁−1)ℎ)⋯𝑀(𝑧+ℎ)⋅𝑀(𝑧). (3.8) It is well known that if ℎ is irrational, and 𝑀 is sufficiently regular in 𝑧, then the limit (3.7) exists for almost all 𝑧 and is independent of 𝑧.

Set ℎ≡2𝜋/𝜀 [2]. Let 𝑀 be the monodromy matrix associated to a consistent basis (𝜓_1,2). Consider the monodromy equation:𝐹_𝑛+1=𝑀(𝑧+𝑛ℎ,𝐸)𝐹_𝑛∀𝑛∈ℤ,𝐹_𝑛∈ℂ². (3.9) The Lyapunov exponent of the monodromy equation (3.9) is defined by 𝜃(𝐸,𝜀)=𝜃(𝑀(𝑧,𝐸),ℎ). (3.10) 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 that 𝜀/2𝜋 is irrational. The Lyapunov exponents Θ(𝐸,𝜀) and 𝜃(𝐸,𝜀) are related by the following relation: 𝜀Θ(𝐸,𝜀)=2𝜋𝜃(𝐸,𝜀). (3.11)

3.4. The Asymptotics of the Monodromy Matrix

As 𝑊 and 𝑉 are real on the real line, we construct a monodromy matrix of the following form:𝑀(𝑧,𝐸)=𝐴(𝑧,𝐸)𝐵(𝑧,𝐸)𝐵(𝑧,𝐸)𝐴(𝑧,𝐸). (3.12) 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 of 𝐴 and 𝐵 in the adiabatic case.

Theorem 3.4. Let 𝐸₀ 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 𝐴=𝐴₊(𝐸,𝜀)𝑒^{−2𝑖𝑄+𝜋𝑧}[1+𝑜(1)],𝐵=𝐵₊(𝐸,𝜀)𝑒^{−2𝑖𝑃+𝜋𝑧}𝑌[1+𝑜(1)],0<Im𝑧<𝜀,𝐴=𝐴₋(𝐸,𝜀)𝑒^{2𝑖𝑄−𝜋𝑧}[1+𝑜(1)],𝐵=𝐵₋(𝐸,𝜀)𝑒^{2𝑖𝑃−𝜋𝑧}𝑌[1+𝑜(1)],−𝜀<Im𝑧<0. (3.13)(3.14) 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 1𝐶<𝑇(𝐸,𝜀)|𝐴_±1(𝐸,𝜀)|<𝐶,𝐶<𝑇(𝐸,𝜀)|𝐵_±(𝐸,𝜀)|<𝐶, (3.15) where 𝑇(𝐸,𝜀) is defined in (2.21).
For 𝑌₁ 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 coefficients 𝐴₊, 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 Theorem 3.1

3.5.1. The Upper Bound

Fix 𝑃=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:|𝐴|,|𝐵|≤𝐶(𝑦₀)⋅𝑇(𝐸)⁻¹𝑒^{2𝜋𝑃𝑦0/𝜀}𝑦,Im𝑧=⁰𝜀,|𝐴|,|𝐵|≤𝐶(𝑦₀)⋅𝑇(𝐸)⁻¹𝑒^{2𝜋𝑃𝑦0/𝜀}𝑦,Im𝑧=−⁰𝜀. (3.16) 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 |𝐴|,|𝐵|≤2𝐶(𝑦₀)𝑇(𝐸)⁻¹exp(2𝜋𝑃𝑦₀𝜀),𝑧∈ℝ. (3.17) This leads to the following upper bound for the Lyapunov exponent for the matrix cocycle generated by 𝑀(𝑧,𝐸):𝜃(𝐸,𝜀)≤log(𝑇(𝐸)⁻¹)+𝐶+2𝜋𝑃𝑦₀𝜀, (3.18) where 𝐶 is a constant independent of 𝐸 and 𝜀. Using (3.11) one gets𝜀Θ(𝐸,𝜀)≤2𝜋log(𝑇(𝐸)⁻¹)+𝜀𝐶+2𝜋𝑃𝑦₀. (3.19)

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. Fix 𝜀₀>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 function 𝑧→𝑀(𝑧,𝜀) is analytic in the strip 𝑆={𝑧∈ℂ;0≤Im𝑧≤𝑦₁/𝜀}; (ii)in the strip 𝑆₁={𝑧∈ℂ;𝑦₀/𝜀≤Im𝑧≤𝑦₁/𝜀}⊂𝑆,𝑀(𝑧,𝜀) admits the representation 𝑀(𝑧,𝜀)=𝜆(𝜀)𝑒^{𝑖2𝜋𝑛0𝑧}⋅(𝑀₀(𝜀)+𝑀₁(𝑧,𝜀)); (3.20) for some constant 𝜆(𝜀), some integer 𝑛₀, and a matrix 𝑀₀(𝜀), all of them independent of 𝑧;(iii) 𝑀(𝑧,𝐸)=(¹⁰_{𝛽(𝜀)𝛼(𝜀)});(iv) there exist constants 𝛽>0 and 𝛼∈(0,1) independent of 𝜀 such that |𝛼(𝜀)|≤𝛼 and |𝛽(𝜀)|≤𝛽;(v)𝑚(𝜀)=sup_𝑧∈𝑆1‖𝑀₁(𝑧,𝜀)‖→0 as 𝜀→0. Then, there exists 𝐶>0 and 𝜀₁>0 (both depending only on 𝑦₀,𝑦₁,𝛼,𝛽, and 𝜀↦𝑚(𝜀)) such that, if 0<𝜀<𝜀₁, one has 𝜃(𝑀(⋅,𝜀),ℎ)>log|𝜆(𝜀)|−𝐶𝑚(𝜀). (3.21)

Proposition 3.6 is used by applying the arguments of [5, 10] to get the lower bound for the Lyapunov exponent.

First for 𝜎=(⁰¹₁₀) we prove that the matrix 𝜎𝑀(𝑧,𝐸)𝜎 completes the assumption of Proposition 3.6.

Let 𝑦₀ 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 that 𝑛=𝑁_𝑁 in (5.16) is even; then the following relations hold:𝑄₋=𝑄₊+1,𝑄₋=𝑃₋+1,𝑄₋=𝑃₊. (3.22) Equation (3.22) is derived from the expression of 𝑄_± and 𝑃_±, given in Section 5.3. 𝜆^±_𝑗,𝛽^±_𝑗 are equals for 2≤𝑗≤𝑁. The first expression of formula (3.22) deduced the relation between 𝛽⁺₁ and 𝛽⁻₁ (see (5.27) and (5.32)), the second one from the relation between 𝜆⁻₁ and 𝛽⁻₁ (see (5.21) and (5.32)), and the last one from the relation between 𝜆⁺₁ and 𝛽⁻₁ (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 [𝜑⁻₁(𝐸),𝜑⁺₁(𝐸)]. See Section 4.4.

For 𝐸∈𝑉₀∩ℝ and 𝑧∈𝑆, Theorem 3.4 implies that 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)), (3.23) where 𝑐(𝐸) is independent of 𝑧 and bounded by a constant uniformly in 𝜀 and 𝐸. So, we have 𝜎⋅𝑀(𝑧,𝐸)⋅𝜎=𝐴₋(𝐸,𝜀)𝑒^{−2𝜋𝑖𝑄−𝑧}⋅(10𝑐(𝐸)0+𝑜(1)). (3.24) This gives that the matrix-valued function 𝑧↦𝜎⋅𝑀(𝑧,𝐸)⋅𝜎 (3.25) satisfies the assumptions of Proposition 3.6.

Remark 3.7. When 𝑛=𝑁_𝑁 in (5.16) is odd, then we get 𝑄₊=𝑄₋+1,𝑃₋=𝑄₊,𝑃₊+1=𝑄₊. (3.26) For 𝐸∈𝑉₀∩ℝ and 𝑧∈𝑆, Theorem 3.4 implies that 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), (3.27) where 𝑐(𝐸) is independent of 𝑧 and bounded by a constant uniformly in 𝜀 and 𝐸. So, we have 𝜎⋅𝑀(𝑧,𝐸)⋅𝜎=𝐴₊(𝐸,𝜀)𝑒^{−2𝜋𝑖𝑄+𝑧}⋅(0𝑐(𝐸)01+𝑜(1)). (3.28) 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 estimates𝜃(𝐸,𝜀)≥log|𝐴₋|+𝑜(1). (3.29) Taking into account (1.2) we get that 𝜀Θ(𝐸,𝜀)≥2𝜋log|𝐴₋|+𝑜(𝜀). (3.30) By (3.15) we get𝜀Θ(𝐸,𝜀)≥2𝜋log(𝑇(𝐸)⁻¹)+𝑂(𝜀). (3.31)

3.5.3. Conclusion

Now we obtain (3.1), by comparing (3.19) and (3.31). Indeed we see that𝜀Θ(𝐸,𝜀)=2𝜋log(𝑇(𝐸)⁻¹)+𝑜(1). (3.32) The expression of 𝑇(𝐸) given by (3.32) and (3.31) gives (3.1) for any 𝐸∈𝑉₀∩ℝ.

Recall that 𝑉₀∩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:−𝑑²𝑑𝑥²𝜓(𝑥)+[𝑉(𝑥)+𝑊(𝜀𝑥+𝜁)]𝜓(𝑥)=𝐸𝜓(𝑥),𝑥∈ℝ. (4.1) 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 𝐿²_loc and that 𝑊 is analytic in a strip in the neighborhood 𝑆_𝑌 of the real line.

The parameter 𝜁 is an auxiliary complex parameter used to decouple the slow variable 𝜁=𝜀𝑥 and the fast variable 𝑥. The idea of this method is to study solutions of (4.1) in some domains of the complex plane of 𝜁 and then to recover information on their behavior in 𝑥∈ℝ. Therefore, for 𝐷 being a complex domain, one studies solutions satisfying the following condition:𝜓(𝑥+1,𝜁)=𝜓(𝑥,𝜁+𝜀),∀𝜁∈𝐷. (4.2) The aim of the WKB method is to construct solutions to (4.1) satisfying (4.2) and that have simple asymptotic behavior when 𝜀 tends to 0. This is possible in certain special domains of the complex plane of 𝜁. These domains will depend continuously on 𝑉,𝑊, and 𝐸. We 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, the canonical 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−𝑑²𝑑𝑥²𝜓(𝑥)+𝑉(𝑥)𝜓(𝑥)=ℰ(𝜁)𝜓(𝑥),ℰ(𝜁)=𝐸−𝑊(𝜁),𝑥∈ℝ, (4.3) that are moreover analytic in 𝜁 on a given regular domain.

Let 𝜁₀ 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 setΨ_±(𝑥,𝜁)=𝑞(ℰ)𝑒^{∫ℰℰ0Ω±}𝜓_±(𝑥,ℰ),𝑞(ℰ)=𝑘^′(ℰ),ℰ=ℰ(𝜁). (4.4) The functions Ψ_± are called the canonical Bloch solutions normalized at the point 𝜁₀.

The properties of the differential Ω imply that the solutions Ψ_± can be analytically continued from 𝑉₀ to any regular domain containing 𝑉₀.

The Wronskian of Ψ_± satisfies (see [13])𝑤(Ψ₊(⋅,𝜁),Ψ₋(⋅,𝜁))=𝑤(Ψ₊(⋅,𝜁₀),Ψ₋(⋅,𝜁₀))=𝑘^′(ℰ₀)𝑤(𝜓₊(⋅,𝜁₀),𝜓₋(⋅,𝜁₀)). (4.5) For ℰ₀∉𝑄∪{𝐸_𝑙}, the Wronskian 𝑤(Ψ₊(⋅,𝜁),Ψ₋(⋅,𝜁)) is nonzero.

4.2. Solutions Having Standard Asymptotic Behavior

Fix 𝐸=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. Let 𝜎∈{+,−}. We say that, in 𝐷, a consistent solution 𝑓 has standard behavior (or standard asymptotics) if (i)there exists 𝑉₀, 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: 𝑓=𝑒^{𝜎(𝑖/𝜀)∫𝜁𝜁0𝜅(𝑢)𝑑𝑢}(Ψ_𝜎+𝑜(1))as𝜀tendsto0; (4.6)(iv) this asymptotic can be differentiated once in 𝑥 without loosing its uniformity properties. We set 𝑓^∗(𝑥,𝜁,𝐸,𝜀)=𝑓(𝑥,𝜁,𝐸,𝜀). (4.7)

We call 𝜁₀ the normalization point for 𝑓. To say that a consistent solution 𝑓 has standard behavior, we will use the following notation: 𝑖𝑓∼exp(𝜎𝜀^𝜁_𝜁0𝜅(𝑢)𝑑𝑢)Ψ_𝜎. (4.8)

4.3. Some Results on the Continuation of Asymptotics

4.3.1. Description of the Stokes Lines near [𝜑⁻₁(𝐸),𝜑⁺₁(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:Im^𝜁_𝜁0(𝜅(𝜉)−𝜅(𝜁₀))𝑑𝜉=0. (4.9) 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 that 𝑊^′(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 𝜅(𝜁)=𝜅(𝜁₀)+𝑐𝜁−𝜁₀(1+𝑜(1)). (4.10) So ^𝜁_𝜁0(𝜅(𝜁)−𝜅(𝜁₀))𝑑𝜁=𝑐(𝜁−𝜁₀)^3/2(1+𝑜(1)). (4.11)

4.3.3. Stokes Lines for 𝐸₀∈𝐽

We describe the Stokes lines beginning at 𝜑⁻₁(𝐸) and 𝜑⁺₁(𝐸). Since 𝑊 is real on ℝ, the set of the Stokes lines is symmetric with respect to the real line.

First, 𝜅₁ is real on the interval [𝜑⁻₁(𝐸),𝜑⁺₁(𝐸)]; therefore this set is a Stokes line starting at 𝜑⁻₁(𝐸). The two other Stokes lines beginning at 𝜑⁻₁(𝐸) are symmetric with respect to the real axis. We denote by 𝜎⁻₁(𝐸) the Stokes line going downward and by 𝜎₁(𝐸) its symmetric. Similarly, we denote by 𝜎⁺₁(𝐸) and 𝜎⁺₁(𝐸) the two other Stokes lines starting at 𝜑⁺₁(𝐸), and 𝜎⁺₁(𝐸) goes upward. These Stokes lines are represented in Figure 5.

Figure 5 

Stokes lines for 𝐸∈𝐽.

Lemma 4.2. The Stokes lines 𝜎⁻₁(𝐸) and 𝜎⁺₁(𝐸) satisfy the following properties. (i)The Stokes lines 𝜎⁻₁(𝐸) and 𝜎⁺₁(𝐸) stay vertical. (ii)𝜎⁻₁(𝐸) and 𝜎⁺₁(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 [𝜑⁻₁(𝐸),𝜑⁺₁(𝐸)]

We recall this result, proved in [17].

Proposition 4.3 (see [17]). Fix 𝐸₀∈𝐽, and let 𝜅₁ be a continuous determination of the complex momentum on [𝜑⁻₁(𝐸),𝜑⁺₁(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: 𝑓₁∼𝑒^{(𝑖/𝜀)∫}𝜅₁Ψ⁺₁,𝑓^∗₁(𝑥,𝜁,𝐸)=𝑓(𝑥,𝜁,𝐸) (4.12)(4.13) to the left of 𝜎⁻₁(𝐸)∪𝜎⁺₁(𝐸) (resp., to the right of 𝜎⁺₁(𝐸)∪𝜎⁻₁(𝐸)). The determination 𝜅₁ is the continuation of 𝜅₁ through {𝜁∈(𝜎⁻₁(𝐸)∪𝜎⁺₁(𝐸))∩𝑆_𝑌;Im𝜅₁(𝜁)>0}(resp., through {𝜁∈(𝜎⁻₁(𝐸)∪𝜎⁺₁(𝐸))∩𝑆_𝑌;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 Theorem 3.4

The Proof of Theorem 3.4 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 solution 𝑓 of (3.2) satisfying (3.3).

To use the complex WKB method, we perform the following change of variable in (3.2):𝑥−𝑧→𝑥,𝜀𝑧→𝜁. (5.1) 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,𝑓^∗₁) constructed around [𝜑⁻₁,𝜑⁺₁] as in Proposition 4.3. Then the monodromy matrix associated to the basis (𝑓₁,𝑓^∗₁) (defined in Section 3.2.1) satisfies 𝑓₁𝑓(𝑥,𝜁,𝐸,𝜀)_∗1𝑓(𝑥,𝜁,𝐸,𝜀)=𝑀(𝜁,𝐸,𝜀)¹𝑓(𝑥,𝜁−2𝜋,𝐸,𝜀)_∗1(𝑥,𝜁−2𝜋,𝐸,𝜀). (5.2) The aim of this section is the computation of 𝑀(𝜁,𝐸,𝜀). The definition of the monodromy matrix implies that𝐴(𝜁)=𝑀_1,1=𝑤(𝑓₁(𝑥+2𝜋,𝜁)𝑓^∗₁(𝑥,𝜁))𝑤(𝑓₁(𝑥,𝜁),𝑓^∗₁(𝑥,𝜁)),𝐵(𝜁)=𝑀₁₂=𝑤(𝑓₁(𝑥,𝜁),𝑓₁(𝑥,𝜁+2𝜋))𝑤(𝑓₁(𝑥,𝜁),𝑓^∗₁.(𝑥,𝜁)) (5.3) This gives that the monodromy matrix is analytic in 𝜁 in the strip 𝑆_𝑌 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∈𝑆_𝑌. This is an immediate consequence of the properties of 𝑓₁.

Therefore, we will compute the Fourier series of 𝐴 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.

Let ℎ and 𝑔 have a standard asymptotic behavior in regular domains 𝐷_ℎ and 𝐷_𝑔 and solutions of (4.1): ℎ∼𝑒^{(𝑖/𝜀)∫𝜁𝜁ℎ𝜅ℎ𝑑𝜁}𝜓_ℎ(𝑥,𝜁),𝑔∼𝑒^{(𝑖/𝜀)∫𝜁𝜁𝑔𝜅𝑔𝑑𝜁}𝜓_𝑔(𝑥,𝜁). (5.4) Here, 𝜅_ℎ (resp., 𝜅_𝑔) is an analytic branch of the complex momentum in 𝐷_ℎ (resp., 𝐷_𝑔), Ψ_ℎ (resp., Ψ_𝑔) is the canonical Bloch solution defined on 𝐷_ℎ (resp., 𝐷_𝑔), and 𝜁_ℎ (resp., 𝜁_𝑔) is the normalization point for ℎ (resp., 𝑔).

As the solutions ℎ and 𝑔 satisfy the consistency condition (4.2), their Wronskian is 𝜀-periodic in 𝜁.

5.1.1. Arcs

We assume that 𝐷_𝑔∩𝐷_ℎ contains a simply connected domain 𝐷. Let 𝛾 be a regular curve going from 𝜁_𝑔 to 𝜁_ℎ in the following way: staying in 𝐷_𝑔, it goes from 𝜁_𝑔 to some point in 𝐷, and then, staying in 𝐷_ℎ, it goes to 𝜁_ℎ. We say that 𝛾 is an arc associated to the triple ℎ,𝑔, and 𝐷.

As 𝐷 is simply connected, all the arcs associated to the triple, ℎ,𝑔, and 𝐷. As 𝐷 is simply connected, all the arcs associated to one and the same triple can naturally be considered as equivalent; we denote them by 𝛾(ℎ,𝑔,𝐷).

We continue 𝜅_ℎ and 𝜅_𝑔 analytically along 𝛾(ℎ,𝑔,𝐷). From the properties of 𝜅, we deduce that, for 𝑉 a small neighborhood of 𝛾, one has𝜅_𝑔(𝜁)=𝜎𝜅_ℎ+2𝜋𝑚,forsome𝑚∈ℤ,𝜎∈{1,−1}. (5.5)𝜎=𝜎,ℎ,𝑔,𝐷 is called the signature of 𝛾(ℎ,𝑔,𝐷), and 𝑚=𝑚(ℎ,𝑔,𝐷) the index of 𝛾(ℎ,𝑔,𝐷).

5.1.2. The Meeting Domain

Let 𝐷 be as above. We call 𝐷 a meeting domain, if, in 𝐷, the functions Im𝜅_ℎ and Im𝜅_𝑔 do not vanish and are of opposite signs.

Note that, for small values of 𝜀, whether 𝜁↦𝑔(𝑥,𝜁) and 𝜁↦ℎ(𝑥,𝜁) increase or decrease is determined by the exponential factor 𝑒^{∫𝜁𝜁𝑔}𝜅_𝑔𝑑𝜁 and 𝑒^{∫𝜁𝜁ℎ}𝜅_ℎ𝑑𝜁. 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 𝑆(ℎ,𝑔,𝐷)=_{𝛾(ℎ,𝑔,}𝐷⁾𝜅_𝑔𝑑𝜁, (5.6) the action of the arc 𝛾=𝛾(ℎ,𝑔,𝐷). Clearly, the action takes the same value for equivalent arcs.

Assume that ℰ(𝜁)∉𝑃∪𝑄 along 𝛾(ℎ,𝑓,𝐷). Consider the function 𝑞_𝑔=𝑘^′(ℰ) and the 1-form Ω_𝑔(ℰ(𝜁)) in the definition of Ψ_𝑔. Continue them analytically along 𝛾. We set 𝑞𝐴(ℎ,𝑔,𝛾)=(^𝑔𝑞_ℎ)|_𝜁=𝜁ℎ𝑒^{∫𝛾𝜔𝑔}. (5.7)𝐴 is called the the amplitude of the arc 𝛾. The properties of Ω imply that the amplitudes of two equivalent arcs 𝛾(ℎ,𝑔,𝐷) coincide.

5.2. Results on the Fourier Coefficients

We recall the following result from [5].

Proposition 5.1. Let 𝑑=𝑑(ℎ,𝑔) be a meeting domain for ℎ and 𝑔, and let 𝑚=𝑚(ℎ,𝑔,𝑑) be the corresponding index. Then 𝑤(ℎ,𝑔)=𝑤_𝑚𝑒^{(2𝜋𝑖𝑚/𝜀)(𝜁−𝜁ℎ)}(1+𝑜(1));𝜁∈𝑆(𝑑), (5.8) where 𝑤_𝑚 is the constant given by 𝑤_𝑚=𝐴(ℎ,𝑔,𝑑)𝑒^{(𝑖/𝜀)𝑆(ℎ,𝑔,𝑑)}𝑤(𝜓₊(⋅,𝜁_ℎ),𝜓₋(⋅,𝜁_ℎ)). (5.9) Here 𝜓₊=𝜓_ℎ 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 Index 𝑚

Let 𝜁₀ 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 branch 𝜅 along the curve 𝛾 of a period (𝛾,𝜅). In a neighborhood of 𝜁₀, the starting point 𝛾, one has 𝜅(𝜁+2𝜋)=𝜎𝜅(𝜁)+2𝜋𝑚,𝜎∈{∓1},𝑚∈ℤ. (5.10) 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 𝒢=∪_𝑘∈ℤ{∪^𝑁_𝑗=1𝒢_𝑗+2𝜋𝑘} (5.11) 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,…,𝑟_𝑛 be the values that Re𝜅 takes consecutively at these intersection points as 𝜁 moves along 𝛾 from 𝜁₀ to 𝜁₀+2𝜋. Then, 𝜎(𝛾,𝜅)=(−1)^𝑛,1𝑚(𝛾,𝜅)=𝜋(𝑟_𝑛−𝑟_𝑛−1+𝑟_𝑛−2−⋯+(−1)^𝑛−1𝑟₁)=(−1)^𝑛−1𝜋(𝑟₁−𝑟₂+⋯+(−1)^𝑛−1𝑟_𝑛). (5.12)

5.3. The Fourier Coefficients

5.3.1. For 𝐵

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 𝐷ℎ(𝑥,𝜁)=𝑓(𝑥,𝜁),𝑔(𝑥,𝜁)=(𝑇𝑓)(𝑥,𝜁),with(𝑇𝑓)(𝑥,𝜁)=𝑓(𝑥,𝜁+2𝜋),_ℎ=𝒟,𝐷_𝑔=𝒟−2𝜋,𝜁_ℎ=𝜁₀,𝜁_𝑔=𝜁₀−2𝜋,∀𝜁∈𝐷_ℎ,𝜅_ℎ(𝜁)=𝜅(𝜁),∀𝜁∈𝐷_𝑔,𝜅_𝑔(𝜁)=𝜅(𝜁+2𝜋). (5.13)(5.14)(5.15) We will start by the following.

5.3.2. Above the Real Line

We take the meeting domain 𝐷₀ as the subdomain of the strip 0<Im𝜁<𝑌 between the Stokes lines 𝜎⁺₁−2𝜋 and 𝜎⁺₁. In this domain we have Im𝜅_𝑔=−Im𝜅_ℎ<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 𝜅_𝑔(𝜁)=𝜅_ℎ(𝜁+2𝜋), and to go from 𝜁 to 𝜁+2𝜋 one has to intersect ℬ₁.

The arc 𝛾₀ connects the point 𝜁_𝑔 to 𝜁_ℎ. By (5.14), this defines the period (𝛾₀,𝜅_𝑔). Using (5.15), one gets that 𝑚(𝑓(⋅,𝜁),𝑓(⋅,𝜁+2𝜋,𝐷₀))=𝑚(𝛾₀+2𝜋,𝜅) [5].

We use Lemma 5.2 to compute the index. To do this, we have to compute Re𝜅 at the intersection of 𝛾₀+2𝜋 and 𝒢.

As 𝜁→Re𝜅(𝜁) is constant on any connected component 𝒢_𝑗 of 𝒢. Let us start by defining the index 𝜆⁺_𝑗 of 𝒢_𝑗, the result of the alternated character of the coefficients (…,+,−,+,−,…) due to the crossing of 𝒢_𝑗 (see Lemma 5.2). We notice that 𝜆⁺_𝑗∈{−1,0,1}.

We set𝑁_𝑗=1+^𝑗−1_𝑖=1(^𝑛𝑖_𝑙=1(𝑜_𝑙−1)). (5.16) Here 𝑛_𝑗 is the number of extremum in 𝒢_𝑗, and 𝑜_𝑙 is the order of the 𝑙th extremum. The following relations hold:𝜆⁺₁=1+(−1)^{∑𝑛1𝑙=1(𝑜𝑙−1)}2, (5.17) and for 2≤𝑗≤𝑁,𝜆⁺_𝑗=(−1)^𝑁𝑗(−1+(−1)^{∑𝑛𝑗𝑙=1(𝑜𝑙−1)}2). (5.18) Without loss of generality we assume that Re𝜅=0 on (0,𝜑⁻₁). By the above notation, we get that𝑚(𝛾₀,𝜅)=(−1)^𝑛(𝑝₁𝜆⁺₁+(𝑝₁+𝑝₂)𝜆⁺₂+⋯+(^𝑗_𝑖=1𝑝_𝑖)𝜆⁺_𝑗+⋯+(^𝑁_𝑖=1𝑝_𝑖)𝜆⁺_𝑁)=𝑃₊. (5.19) Here 𝑝_𝑖 is the index of ℬ_𝑖 and 𝑛=𝑁_𝑁.

Now using (5.3) for 𝐵 and Proposition 5.1 we get𝐵=𝐴(𝑓,𝑇(𝑓),𝐷₀)𝑒^{(𝑖/𝜀)𝑆(𝑓,𝑇(𝑓),}𝐷₀₎⋅𝑒^{−2𝑖𝜋𝑃+𝜁0/𝜀}=𝐵₊(𝐸,𝜀)⋅𝑒^{−(𝑖/2𝜋𝜀)𝑃+𝜁0},𝑇(𝑓)(𝑥,𝜁)=𝑓(𝑥,𝜁+2𝜋). (5.20)

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 𝜎⁻₁−2𝜋 and 𝜎⁻₁ as a regular domain, which we denote by 𝐷₀. We set 𝛾₀=𝛾(𝑓,𝑇(𝑓),𝐷₁); then it defines a period, and so 𝑚(𝑓,𝑇(𝑓),𝐷₁)=𝑚(𝛾₀+2𝜋,𝜅). The curve defining a period equivalent to (𝛾₀+2𝜋,𝜅) is represented in Figure 6.

Figure 6 

Periods equivalent to 𝛾0 and 𝛾1.

The computation of the index 𝜆⁻_𝑗 of 𝒢_𝑗 gives in this case that𝜆⁻₁=−1+(−1)^{∑𝑛1𝑖=1(𝑜𝑙−1)}2,𝜆⁻_𝑗=(−1)^𝑁𝑗(−1+(−1)^{∑𝑗𝑙=𝑛𝑙=1(𝑜𝑙−1)}2). (5.21)(5.22) By this notation we get that𝑚(𝛾₀,𝜅)=(−1)^𝑛(𝑝₁𝜆⁻₁+(𝑝₁+𝑝₂)𝜆⁻₂+⋯+(^𝑗_𝑖=1𝑝_𝑖)𝜆⁻_𝑗+⋯+(^𝑁_𝑖=1𝑝_𝑖)𝜆⁻_𝑁)=𝑃₋. (5.23) Using (5.3) for 𝐵 and Proposition 5.1 we get𝐵_𝑃−=𝐴(𝑓,𝑇(𝑓),𝐷₁)𝑒^{(𝑖/𝜀)(𝑆(𝑓,𝑇𝑓,}𝐷_{1)−2𝜋𝑃−𝜁0)}=𝐵₋(𝐸,𝜀)⋅𝑒^{(𝑖/𝜀)2𝜋𝑃−𝜁0},(𝑇𝑓)(𝑥,𝜁)=𝑓(𝑥,𝜁+2𝜋). (5.24)

5.4. For 𝐴

For the computation of 𝐴 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 ℎ(𝑥,𝜁)=𝑓^∗𝐷(𝑥,𝜁),𝑔(𝑥,𝜁)=(𝑇𝑓)(𝑥,𝜁),with(𝑇𝑓)(𝑥,𝜁)=𝑓(𝑥,𝜁+2𝜋),_ℎ=𝒟^∗,𝐷_𝑔=𝒟−2𝜋,𝜁_ℎ=𝜁₀,𝜁_𝑔=𝜁₀−2𝜋,∀𝜁∈𝐷_ℎ,𝜅_ℎ(𝜁)=−𝜅(𝜁),∀𝜁∈𝐷_𝑔,𝜅_𝑔(𝜁)=𝜅(𝜁+2𝜋). (5.25)(5.26)

5.4.1. Above the Real Line

In this case, 𝐷₀, the meeting domain, is the subdomain of the strip {0<Im𝜁<𝑌} located between the lines 𝜎⁺₁−2𝜋 and 𝜎⁻₁, the symmetric to 𝜎⁻₁ with respect to ℝ. The arc 𝛾(𝑓^∗,𝑇(𝑓),𝐷₀) defines a period (𝛾₀,𝜅_𝑔); in Figure 7 we represent the curve 𝛾₀+2𝜋.

Figure 7 

Periods equivalent to 𝛾0 and 𝛾1.

Similarly to the computation of 𝐵, we define the index 𝛽⁺_𝑗 of 𝒢_𝑗, and in this case, we have 𝛽⁺₁=−1+(−1)^{∑𝑛1𝑖=1(𝑜𝑙−1)}2,𝛽⁺_𝑗=(−1)^𝑁𝑗(−1+(−1)^{∑𝑗𝑙=𝑛𝑙=1(𝑜𝑙−1)}2). (5.27)(5.28) By this notation we get that𝑚(𝛾₀,𝜅)=(−1)^𝑛(𝑝₁𝛽⁺₁+(𝑝₁+𝑝₂)𝛽⁺₂+⋯+(^𝑗_𝑖=1𝑝_𝑖)𝛽⁺_𝑗+⋯+(^𝑁_𝑖=1𝑝_𝑖)𝛽⁺_𝑁)=𝑄₊. (5.29) Using (5.3) for 𝐴 and Proposition 5.1 we get that 𝑚(𝑓^∗,𝑇𝑓,𝐷₀)=𝑚(𝛾₀,𝜅)=𝑄₊,𝐴=𝐴(𝑓^∗,𝑇𝑓,𝐷₀)𝑒^{(𝑖/𝜀)(𝑆(𝑓∗,𝑇𝑓,}𝐷_{0)−2𝜋𝑄+𝜁0)}=𝐴₊(𝐸,𝜀)⋅𝑒^{−(𝑖/𝜀)2𝜋𝑄+},(𝑇𝑓)(𝑥,𝜁)=𝑓(𝑥,𝜁+2𝜋). (5.30)(5.31)

5.4.2. Below the Real Line

In this case, 𝐷₁, the meeting domain, is the subdomain of the band {−𝑌<𝐼𝑚𝜁<0} located between the lines 𝜎⁺₁, symmetric of 𝜎⁺₁ with respect to ℝ, and 𝜎⁻₁−2𝜋. The arc 𝛾(ℎ,𝑔,𝐷₁) defines a period (𝛾₁,𝜅_𝑔); the curve 𝛾₁+2𝜋 is represented in Figure 7. One obtains that𝛽⁻₁=1+(−1)^{∑𝑛1𝑙=1(𝑜𝑙−1)}2, (5.32) and for 2≤𝑗≤𝑁,𝛽⁻_𝑗=(−1)^𝑁𝑗(−1+(−1)^{∑𝑛𝑗𝑙=1(𝑜𝑙−1)}2). (5.33) So by the above notation we get that𝑚(𝛾₀,𝜅)=(−1)^𝑛(𝑝₁𝛽⁻₁+(𝑝₁+𝑝₂)𝛽⁻₂+⋯+(^𝑗_𝑖=1𝑝_𝑖)𝛽⁻_𝑗+⋯+(^𝑁_𝑖=1𝑝_𝑖)𝛽⁻_𝑁)=𝑄₋. (5.34) One obtains that 𝑚(𝑓^∗,𝑇(𝑓),𝐷₁)=𝑚(𝛾₁+2𝜋,𝜅)=𝑄₋, and𝐴=𝐴(𝑓^∗,𝑇𝑓,𝐷₁)𝑒^{(𝑖/𝜀)(𝑆(𝑓∗,𝑇𝑓,}𝐷_{1)−2𝜋𝑄−𝜁0)}=𝐴₋(𝐸,𝜀)⋅𝑒^{(𝑖/𝜀)2𝜋𝑄−𝜁0},(𝑇𝑓)(𝑥,𝜁)=𝑓(𝑥,𝜁+2𝜋). (5.35)

5.4.3. The Proof of (3.15)

Let 𝑔∈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 𝛾_𝑔 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]). Let 𝐸∈𝐽. If 𝛾_𝑔 is positively oriented, then 𝑆(𝑔)=±2_𝑔±𝑖0Im𝜅𝑑𝜁. (5.36) Here, in the left-hand side, one integrates in the increasing direction on the real axis.

Below we give details only for 𝐴₊; the same way could be used to obtain the result for 𝐴₋ and 𝐵−±.

We recall that the definition of the amplitude of an arc 𝛾 is given in (5.7) and the coefficient 𝐴₊ is defined in (5.31). This definition implies that 𝐴(𝑓^∗,𝑇(𝑓),𝐷₀) 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 𝐶₁<|𝐴(𝑓^∗,𝑇(𝑓),𝐷₀)|<𝐶₂. (5.37) Let 𝛾=𝛾(𝑓^∗,𝑇(𝑓),𝐷₀) 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 has𝑖|exp(𝜀𝑆(𝑓^∗,𝑇(𝑓),𝐷₀1))|=|exp(−𝜀_𝛾Im𝜅_𝑔1𝑑𝜁)|=exp(−𝜀_{𝑔∈𝒢𝑔}Im𝜅_𝑔𝑑𝜁). (5.38)Im𝜅_𝑔<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 𝑖|exp(𝜀𝑆((𝑓^∗,𝑇(𝑓),𝐷₀)))|=_𝑔∈𝒢(𝑡(𝑔))⁻¹. (5.39) This ends the proof of (3.15) for 𝐴₊.

Acknowledgments

The authors would like to thank F. Klopp for proposing this problem and many valuable comments and remarks and the referee for his valuable comments which improve the paper. M. Marx thanks M. Ben Ammar the director of the Research Unity 01/UR/15-01 for the financial support by paying her mission in Tunisia. H. Najar thanks R. Assel and M. Dimassi for suggesting some interesting references and for their stimulating discussion. Researches are supported by CMCU 09G1504 and the research Unity 01/UR/15-01 projects.