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: where is -periodic, is -periodic, and small is chosen so that the potential 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 see [2]. The operator can be regarded as an adiabatic perturbation of the periodic operator : Equation (1.1) is one of the main models of solid state physics. The function is the wave function of an electron in a crystal with an external electric field. represents the potential of the perfect crystal; as such it is periodic. The potential 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 be the dispersion relation associated to . Consider the complex and real-isoenergy curves   and defined by Notice that the iso-energy curves and are -periodic in and and is the Riemann surface uniformizing .

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 can be interpreted as a “classical” Hamiltonian corresponding to (1.1).

In the case when the interval 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 ; loops are represented by horizontal closed curves. In this case the connected components are extended in -direction and bounded in -direction.

In the case when the interval 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 the vertical loops are situated in . In this case the connected components are bounded in direction and extended in -direction.

When 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 , 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. and are periodic: is real-valued and locally square-integrable. is real analytic in the strip .

We define We define the spectral window by

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 acting on 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 for , such that and . The points are the eigenvalues of the self-adjoint operator obtained by considering defined by (1.2) and acting in with periodic boundary conditions (see [6, 7]). The intervals are the spectral bands, and the intervals the spectral gaps. When one says that the th gap is open; when is separated from the rest of the spectrum by open gaps, the th 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 is a real compact interval such that, for all , the window contains exactly isolated bands of the periodic operator. So, we fix two positive integers and and assume that: The bands are isolated. For all these bands are contained in the interior of . :For all , the rest of the spectrum of is outside of .

Remark 1.1. The geometric assumption assures that the iso-energy curve 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 be a real compact interval. We assume that are satisfied. For sufficiently small, for almost all one has Here is the absolutely continuous spectrum of the family of .

Remark 1.3. It is proved in [8] that for one has the following for all and for any , compact, there exists such that for all sufficiently small and all , So, for an interval as in Theorem 1.2, we have
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 is a 1-periodic, real-valued, - function. We recall known facts needed on the present paper and we introduce notations. Basic references are [6, 1012].

2.1. Geometric Description
2.1.1. The Set

As , the set coincides with . It is -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 that do not cross again the real line.

Consider an extremum of of order on the real line, say . Near , the set consists of a real segment and of complex curves symmetric with respect to the real axis and intersecting the real axis only on . The angle between two neighboring curves is equal to Let We set We assume that is so small that (i) is contained in the domain of analyticity of (ii)the set consists of the real line and of the complex lines passing through the real extrema of .

An example of subset is shown in Figure 3.

2.1.2. Notations and Description of

For all , we write with the following properties. (i)(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 is associated to a connected component of which is an unbounded vertical curve. See Remark 2.3. (iii)We generally define the following We set  =  =  =  and  =   =  Let be the extrema of in . We recall that is the order of .

We have the following description.

Lemma 2.1. Fix a compact interval of .
There exists a finite number of real extrema of in . If , there exists such that
For one denotes by the real extrema of in . There exists and a sequence of disjoint and strictly vertical lines of starting at such that

2.2. Bloch Solutions

Let be a solution of the equation satisfying the relation for all and some nonvanishing complex number independent of . Such a solution exists and is called the Bloch solution and is called Floquet multiplier. We discuss its analytic properties as a function of .

As in Section 1.1.2, we denote the spectral bands of the periodic Schrödinger operator by Consider two copies of the complex plane cut along the spectral bands. Paste them together to get a Riemann surface with square root branch points. We denote this Riemann surface by .

One can construct a Bloch solution meromorphic on It is normalized by the condition 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 or on The position of the pole is independent of

For , we denote by the point on different from and having the same projection on as We let The function is another Bloch solution of (2.5). Except at the edges of the spectrum, the functions and are linearly independent solutions of (2.5). In the spectral gaps, and are real-valued functions of , and, on the spectral bands, they differ only by complex conjugation.

2.3. The Bloch Quasimomentum

Consider the Bloch solution The corresponding Floquet multiplier is analytic on Represent it in the form The function is the Bloch Quasimomentum of Its inverse is the dispersion relation of A branching point is a point where .

Let be a simply connected domain containing no branch point of the Bloch Quasimomentum. In one can fix an analytic single-valued branch of say All the other single-valued branches of that are analytic in are related to by the following formulae: Consider the upper half plane of the complex plane. On 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 and for We call this branch the main branch of the Bloch Quasimomentum and denote it by

The function conformally maps onto the first quadrant of the complex plane cut at compact vertical slits starting at the points . It is monotonically increasing along the spectral zones so that the th spectral band, is mapped on the interval Along any open gap, is constant, and is positive and has only one nondegenerate maximum.

Consider , the complex plane cut along the spectral the real line from to . In Figure 4, we drew two curves in and their images under the transformation

All the branch points of are of square root type. Let be a branch point of . In a sufficiently small neighborhood of , the function is analytic in , and Finally, we note that the main branch can be continued analytically to the complex plane cut along and the spectral gaps of the periodic operator

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 , there is a meromorphic function having the following properties: (i)the differential is meromorphic; its poles are the points of where is the set of poles of and is the set of zeros of (ii)all the poles of are simple; (iii)if the residue of at a point is denoted by , one has (iv) if projects into a gap, then (v)if projects inside a band, then

2.4.1. The Complex Momentum

It is the main analytic object of the complex WKB method. Let . We define , in the domain of analyticity of by Here, is the Bloch Quasimomentum defined in Section 2.3. Though depends on , we omit the -dependence. Relation (2.11) translates the properties of into properties of . Hence, is a multivalued analytic function, and its branch points are related to the branch points of the Quasimomentum by the relations Let be a branch point of . If , then is a branch point of square root type.

If is a simply connected set containing no branch points of , we call it regular. Let be a branch of the complex momentum analytic in a regular domain . All the other branches that are analytic in are described by the following formulae: Here and are indexing the branches.

2.4.2. Index of an Interval

Fix . Fix a continuous branch of the complex momentum on We define is the index of associated to .

Let us give some properties of .

Lemma 2.2. Assume that is satisfied. The indices have the following properties. (1)For (2)

Proof. The points and are the ends of a band of : they are distinct or coincide. If they coincide, that is, if the index satisfies Else, we consider the branch of the Quasimomentum associated to . We have that see (2.8), and .
Let us prove point As for any we have We write For and are the ends of a same gap and By periodicity, and are the ends of the same gap.
This ends the proof of Lemma 2.2.

If we say that we cross a band. In this case, and the associated connected component of the iso-energy curve is unbounded vertically.

We notice that , and thus .

Remark 2.3. We can choose the determination of such that

2.4.3. Tunneling Coefficients

For , we denote by a smooth closed curve that goes once around . Notice that this curve is the projection of a closed curve on the complex Riemann surface . We consider the tunneling actions given by It is straightforward to prove that for each of these actions is real and nonzero and that is analytic in a complex neighborhood of (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 is called the tunneling action; we choose the branch so that on is positive. We define So we get being positive we get that 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 are satisfied. Then, on the interval for sufficiently small irrational the Lyapunov exponent of (1.1) is positive and satisfies the asymptotics

This theorem implies that if is sufficiently small and irrational, then, the Lyapunov exponent is positive for all

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 . Consider the family of differential equations indexed by :

Definition 3.2. We say that is a consistent basis of solutions to (3.2) if the two functions are a basis of solutions to (3.2) whose Wronskian is independent of and that are 1-periodic in that is, that satisfy

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

The functions being also solutions of (3.2), we get the relation where (i)(ii) is a -matrix with coefficients independent of .

The matrix is called the monodromy matrix associated to the consistent basis . We recall the following properties of this matrix: The Matrix belongs to which is known to be isomorph to .

3.3. The Lyapunov Exponents and the Monodromy Equation

Consider now a 1-periodic, -valued function, say, and irrational. Consider the finite difference equation: 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 repeats the behavior of solutions of the monodromy equation for 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

The Lyapunov exponent of the finite difference equation (3.6) is where the matrix cocycle is defined as It is well known that if is irrational, and is sufficiently regular in then the limit (3.7) exists for almost all and is independent of

Set [2]. Let be the monodromy matrix associated to a consistent basis . Consider the monodromy equation: The Lyapunov exponent of the monodromy equation (3.9) is defined by 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 is the Lyapunov exponent of (1.1). We have the following result.

Theorem 3.3 (see [5]). Assume that is irrational. The Lyapunov exponents and are related by the following relation:

3.4. The Asymptotics of the Monodromy Matrix

As and are real on the real line, we construct a monodromy matrix of the following form: To get (3.12), it suffices to consider a basis of solutions of the form . The details on the existence and the construction of such a basis are developed in [13].

The following result gives the asymptotics of and in the adiabatic case.

Theorem 3.4. Let be in There exists and a neighborhood of such that, for sufficiently small the family of (3.2) has a consistent basis of solutions for which the corresponding monodromy matrix is analytic in and has the form (3.12). When tends to the coefficients and admit the asymptotics The integers and are specified in Section 3. There exists a constant such that for sufficiently small and one has where is defined in (2.21).
For and such that , there exists a neighborhood of such that the asymptotics (3.13) and (3.14) for and are uniform in

Remark 3.5. The coefficients , and are the leading terms of the asymptotics of the th and th Fourier coefficients of the monodromy matrix coefficients. From Theorem 3.4, one deduces that, in the strip , 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 . The asymptotics (3.13) and (3.14) and estimates (3.15) imply the following estimates for the coefficients of , the monodromy matrix: Here, is a positive constant independent of and The estimates are valid for sufficiently small We recall that is analytic and -periodic in . Equation (3.16) and the maximum principle imply that This leads to the following upper bound for the Lyapunov exponent for the matrix cocycle generated by : where is a constant independent of and Using (3.11) one gets

3.5.2. The Lower Bound

For a family of -valued, -periodic functions of and an irrational number, we recall the following result obtained in [8].

Proposition 3.6. Fix . Assume that there exist and such that and such that, for any one has (i)the function is analytic in the strip ; (ii)in the strip admits the representation for some constant , some integer and a matrix , all of them independent of (iii)(iv) there exist constants and independent of such that and (v) as Then, there exists and (both depending only on and such that, if , one has

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

First for we prove that the matrix completes the assumption of Proposition 3.6.

Let and be fixed such that . The asymptotics of the monodromy matrix are uniform for in and

Let us assume that in (5.16) is even; then the following relations hold: Equation (3.22) is derived from the expression of and given in Section 5.3. are equals for The first expression of formula (3.22) deduced the relation between and (see (5.27) and (5.32)), the second one from the relation between and (see (5.21) and (5.32)), and the last one from the relation between and (see (5.17) and (5.32)). We notice that (3.22) depends only on the indices of because we constructed the consistent basis near . See Section 4.4.

For and Theorem 3.4 implies that where is independent of and bounded by a constant uniformly in and So, we have This gives that the matrix-valued function satisfies the assumptions of Proposition 3.6.

Remark 3.7. When in (5.16) is odd, then we get For and Theorem 3.4 implies that where is independent of and bounded by a constant uniformly in and So, we have The most important properties of the matrix in Proposition 3.6 are that the bigger eigenvalue is [8].

Using (3.21), we get that the Lyapunov exponent of the matrix cocycle associated to satisfies the estimates Taking into account (1.2) we get that By (3.15) we get

3.5.3. Conclusion

Now we obtain (3.1), by comparing (3.19) and (3.31). Indeed we see that The expression of given by (3.32) and (3.31) gives (3.1) for any

Recall that is an open interval containing . The above construction can be carried out for any The end of the proof of Theorem 1.2 follows from the compactness of the interval

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: Here, is 1-periodic and real valued, is a small positive parameter, and the energy is complex; one assumes that is and that is analytic in a strip in the neighborhood of the real line.

The parameter is an auxiliary complex parameter used to decouple the slow variable and the fast variable . The idea of this method is to study solutions of (4.1) in some domains of the complex plane of and then to recover information on their behavior in . Therefore, for being a complex domain, one studies solutions satisfying the following condition: The aim of the WKB method is to construct solutions to (4.1) satisfying (4.2) and that have simple asymptotic behavior when tends to . This is possible in certain special domains of the complex plane of . These domains will depend continuously on and . We will use these solutions to compute the monodromy matrix; we consider and as fixed and construct the WKB objects and solutions in a uniform way for energies near .

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 that are moreover analytic in on a given regular domain.

Let be a regular point (i.e., is not a branch point of ). Let . Assume that Let be a sufficiently small neighborhood of , and let be a neighborhood of such that . In , we fix a branch of the function and consider , the two branches of the Bloch solution and , and the corresponding branches of (see Section 2.4.). For , we set The functions are called the canonical Bloch solutions normalized at the point .

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

The Wronskian of satisfies (see [13]) For , the Wronskian is nonzero.

4.2. Solutions Having Standard Asymptotic Behavior

Fix . Let be a regular domain (i.e., , and simply connected set containing no branch points of ). Fix so that . Let be a continuous branch of the complex momentum in and let be the canonical Bloch solutions normalized at defined on and indexed so that is the Quasimomentum for

Definition 4.1. Let . We say that, in , a consistent solution has standard behavior (or standard asymptotics) if (i)there exists , a complex neighborhood of , and such that is defined and satisfies (4.1) and (4.2) for any ; (ii) is analytic in and in ; (iii)for any compact set , there exists , a neighborhood of such that, for , has the uniform asymptotic: (iv) this asymptotic can be differentiated once in without loosing its uniformity properties. We set

We call the normalization point for To say that a consistent solution has standard behavior, we will use the following notation:

4.3. Some Results on the Continuation of Asymptotics
4.3.1. Description of the Stokes Lines near

This section is devoted to the description of the Stokes lines under assumption

4.3.2. Definition

The definition of the Stokes lines is fairly standard [13, 15]. The integral has the same branch points as the complex momentum. Let be one of them. Consider the curves beginning at and described by the following equation: These curves are the Stokes lines beginning at According to (2.8), the Stokes line definition is independent of the choice of the branch of .

Assume that Equation (2.9) implies that there are exactly three Stokes lines beginning at . The angle between any two of them at this point is equal to . Indeed for near , we have So

4.3.3. Stokes Lines for

We describe the Stokes lines beginning at and . Since is real on , the set of the Stokes lines is symmetric with respect to the real line.

First, is real on the interval ; therefore this set is a Stokes line starting at . The two other Stokes lines beginning at are symmetric with respect to the real axis. We denote by the Stokes line going downward and by its symmetric. Similarly, we denote by and the two other Stokes lines starting at , and goes upward. These Stokes lines are represented in Figure 5.

Lemma 4.2. The Stokes lines and satisfy the following properties. (i)The Stokes lines and stay vertical. (ii) and 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

We recall this result, proved in [17].

Proposition 4.3 (see [17]). Fix and let be a continuous determination of the complex momentum on . There exists a real number a complex neighborhood of and a consistent basis of solutions of (4.1) such that has the standard asymptotic behavior: to the left of (resp., to the right of ). The determination is the continuation of through resp., through .

We mimic the analysis done in [5, Section  5]. Precisely, we start by a local construction of the solution 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 are satisfied.

In the present section, we construct and study a solution of (3.2) satisfying (3.3).

To use the complex WKB method, we perform the following change of variable in (3.2): Then (3.2) takes the form (4.1). In the new variables, the consistency condition (3.3) becomes (4.2). Note also that in the new variables, for two solutions to (4.1) to form a consistent basis, in addition to being a basis of consistent solutions, their Wronskian has to be independent of .

Consider the basis constructed around as in Proposition 4.3. Then the monodromy matrix associated to the basis (defined in Section 3.2.1) satisfies The aim of this section is the computation of The definition of the monodromy matrix implies that This gives that the monodromy matrix is analytic in in the strip and in in a constant neighborhood of . By the definition of we get that and are -periodic in . This is an immediate consequence of the properties of .

Therefore, we will compute the Fourier series of and . 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 and have a standard asymptotic behavior in regular domains and and solutions of (4.1): Here, (resp., ) is an analytic branch of the complex momentum in (resp., ), (resp., ) is the canonical Bloch solution defined on (resp., ), and (resp., ) is the normalization point for (resp., ).

As the solutions and satisfy the consistency condition (4.2), their Wronskian is -periodic in

5.1.1. Arcs

We assume that contains a simply connected domain . Let be a regular curve going from to in the following way: staying in , it goes from to some point in , and then, staying in , it goes to . We say that is an arc associated to the triple and .

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

We continue and analytically along . From the properties of , we deduce that, for a small neighborhood of , one has is called the signature of , and the index of .

5.1.2. The Meeting Domain

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

Note that, for small values of whether and increase or decrease is determined by the exponential factor and . So, roughly, in a meeting domain, along the lines , the solutions and increase in opposite directions.

5.1.3. The Action and the Amplitude of an Arc

We call the integral the action of the arc . Clearly, the action takes the same value for equivalent arcs.

Assume that along . Consider the function and the -form in the definition of . Continue them analytically along We set is called the the amplitude of the arc The properties of imply that the amplitudes of two equivalent arcs coincide.

5.2. Results on the Fourier Coefficients

We recall the following result from [5].

Proposition 5.1. Let be a meeting domain for and , and let be the corresponding index. Then where is the constant given by Here and is complementary to . The asymptotic (5.8) is uniform in and when stays in a fixed compact of and in a small enough neighborhood of .

5.2.1. The Index

Let be a regular point. Consider a regular curve going from to . Let be a branch of the complex momentum that is continuous on . We call the couple a period. Let and be two periods. Assume that one can continuously deform into without intersecting any branching point. By this we define an analytic continuation of to . If the analytic continuation coincides with , we say that the periods are equivalent.

Consider the branch along the curve of a period . In a neighborhood of , the starting point , one has The numbers and are called, respectively, the signature and the index of the period . They coincide for equivalent periods.

Recall that is the preimage with respect to of the union of the spectral gaps of . One has the following.

Lemma 5.2 (see [10]). Let be a period such that starts at a point Assume that intersects exactly times () and that at all intersection points, Let be the values that takes consecutively at these intersection points as moves along from to Then,

5.3. The Fourier Coefficients
5.3.1. For

By (5.3), we have to compute . With this aim in view, we apply the construction done in Section 5.1 with We will start by the following.

5.3.2. Above the Real Line

We take the meeting domain as the subdomain of the strip between the Stokes lines and In this domain we have . Indeed we notice that the sign of