Bounded solutions of self-adjoint second order linear di erence equations with periodic coe cients

In this work we obtain easy characterizations for the boundedness of the solutions of the discrete, self–adjoint, second order and linear unidimensional equations with periodic coe cients, including the analysis of the so-called discreteMathieu equations as particular cases.


Introduction
Discrete Schrödinger operators over nite or in nite paths have been subject of an intensive research over the last four decades. They represent the discrete analogs of one-dimensional self-adjoint operators on a bounded or unbounded interval on the real line, see for instance [1]. In addition, those operators are in relation with Jacobi matrices and hence with the classical theory of orthogonal polynomials.
The particular case of the so-called almost Mathieu operator has deserved special attention not only by its connections to physics but for its rich spectral theory. In fact, one of the main problems in this area, related to the topological structure of the spectra and popularized as the Ten Martini Problem, has been recently solved by concatenating the work of many outstanding researchers, see [2][3][4]. The problem is closely related to the determination of those energies for which the corresponding Schrödinger equation has non trivial bounded eigenfunctions.
The aim of this communication is by far much more modest. We use recent advances in the study of linear di erence equations with periodic coe cients, see [5], to provide easy characterizations for the boundedness of the solutions of the Mathieu equations, that correspond to some speci c Schrödinger equations with periodic potential, see [6]. Moreover, we also extend the results to general second order linear di erence equations with periodic coe cients.

Preliminaries
Throughout the paper, (Z) denotes the vector space of real sequences; that is, (Z) = z∶ Z → R , whereas * (Z) is the set of sequences z ∈ (Z) such that z(k) = for all k ∈ Z. The null sequence, also called the trivial sequence, is denoted by 0. Given z ∈ (Z) and p ∈ N * , for any m ∈ Z we denote by z p,m ∈ (Z) the subsequence of z de ned as Clearly, any sequence z ∈ (Z) is completely determined by the values of the sequences z p,j , for ≤ j ≤ p − . In particular, z , = z, whereas z , and z , are the subsequences of z formed by the even or odd indexes, respectively. Moreover, the sequences z ,m are the shift subsequences of z, since z ,m (k) = z(k + m) for any k ∈ Z. Notice that if we also allow p = − , then z − ,m are the ipped shift subsequences of z, The sequence z ∈ (Z) is called periodic with period p ∈ N * if it satis es that which also implies that z(kp + m) = z(m) for any k, m ∈ Z.
The set of periodic sequences with period p is denoted by (Z; p) and we de ne * (Z; p) = (Z; p)∩ * (Z). In particular (Z; ) consists of all constant sequences and then, it is identi ed with R. Lemma 2.1. Given z ∈ (Z), then z is bounded i there exists p ∈ N * such that z p,j is bounded, for ≤ j ≤ p − and then, z r,m is bounded for any r ∈ N * and any m ∈ Z.
Given p ∈ N * , then z ∈ (Z; p) i z p,m ∈ (Z; ) for any m ∈ Z. Moreover, all periodic sequence is also bounded.
Given p ∈ N * , a ∈ (Z; p) and c ∈ * (Z; p), consider the associated self-adjoint operator ∆ a,c ∶ (Z) → (Z), de ned as and the corresponding (irreductible) homogeneous equation The sequences a and c are called the coe cients of the Equation (2) and any sequence z ∈ (Z) satisfying the Identity (2) is called a solution of the equation. It is well-known that for any z , z ∈ R and any m ∈ Z, there exists a unique solution of Equation (2) satisfying z(m) = z and z(m + ) = z . The problem we are interested in, can be formulated as follows: For which coe cients a, c ∈ (Z; p) has the equation ∆ a,c (z) = 0 bounded solutions, other than the trivial one?
Operator (1), and hence Equation (2), encompasses many speci c examples that have been widely considered in the literature. For instance, when c(k) > for any k ∈ Z, then ∆ a,c = −L q , the Schrödinger operator on the in nite path with conductance c and potencial q(k) = a(k) − c(k) − c(k − ); that is, In particular, when c(k) = , for any k ∈ Z, then ∆ a,c is known as the Harper operator and denoted by H a . More speci cally when, in addition, the coe cient a is given by a(k) = E − λ cos( πωk + θ), k ∈ Z, then the operator H a is called Mathieu operator and the parameters E, λ ∈ R, ω ∈ Q, θ ∈ [ , π), are called the energy, coupling, frequency, and phase, respectively. In this case the operator H a is usually represented as H E,λ,ω,θ . If we permit the frequency not to be a rational number; that is, ω ∈ R, then H E,λ,ω,θ is called almost Mathieu operator, but it does not have periodic coe cients. Therefore, in this work we are only interested in Mathieu operators; that is, in rational frequencies. We must bear in mind that when ω = m p , where m ∈ Z and p ∈ N * are relative primes, then a ∈ (Z; p).
The interested reader can nd the physics meaning of these parameters and the physics background of these kind of operators in [1][2][3] and also in [7].
The paper [5] was devoted to the Floquet Theory for the equation ∆ a,c = 0; that is, to the condition under which the above equation has periodic solutions. Since any periodic solution is bounded, this characterization gives us only a partial answer to the main question. However, we can follow the same techniques as in [5] to completely solve the question.
We end this preliminary section by remarking that when only a nite interval in Z is considered, namely when k = , , . . . , n for some n ∈ N, then Equation (2) must be supplied with some boundary conditions and it is related with the inversion of nite and symmetric Jacobi matrices, see for instance [8]. Another interesting application of these boundary value problems falls in the ambit of Organic Chemistry, see Examples 1 and 2 in page 364 of [5]. In this case, all the eigenfuncions are bounded, so the main problem is nothing else that the consideration of the eigenvalue problem. For the Mathieu equation with null frequency, this analysis in the nite interval case can be found in [9].

The easiest case
The most simple case of the proposed problem corresponds to a, c ∈ (Z; ); that is, when the coe cients of ∆ a,c are constant; i.e. a ∈ R and c ∈ R * . Self-adjoint linear di erence equations with constant coe cients can be characterized as those satisfying that z ∈ (Z) is a solution i any shift and any ipped shift of z is also a solution. Moreover, in this case, Equation (2) is equivalent, in the sense that both have the same solutions, to the Chebyshev equation with parameter q where q = a c . So, we can say that the most simple case to analyze corresponds to both the uncoupled Harper equation and the coupled Harper equation with null frequency. Moreover, these two kinds of equations can be viewed in an uni ed manner as Chebyshev equations. Any solution of a Chebyshev equation with parameter q is called Chebyshev sequence with parameter q.
Recall that a polynomial sequence is a sequence of Chebyshev polynomials if it satis es the following three-term recurrence, see [10], Therefore, any Chebyshev sequence with parameter q is of the form {P k (q)} k∈Z , where {P k (x)} k∈Z is a sequence of Chebyshev polynomials. So, many properties of Chebyshev sequences are the consequence of properties of Chebyshev polynomials and conversely. As a by-product of the Proposition 2.1 in [5], we have the following basic result about periodic and bounded Chebyshev sequences.

The general case
Back to the general case, consider p ∈ N * , a ∈ (Z; p), c ∈ * (Z; p) and the associate self-adjoint operator ∆ a,c . Although this scenario seems to be far away from the easiest one analyzed in the previous section, we will show that in fact Chebyshev equations contain all the information needed to conclude the existence of bounded solutions for the di erence equation ∆ a,c (z) = 0. This is true because the main result in [5] establishes that (irreductible) second order di erence equations (not necessarily self-adjoint) with periodic coe cients are basically equivalent to some Chebyshev equation. For the setting concerning to this paper we have the following facts.

Lemma 4.1 ([5, Theorem 3.3]). Given p ∈ N * , a ∈ (Z; p) and c ∈ * (Z; p); there exists q(a, c; p) ∈ R, depending only on the coe cients a and c and on the period p, such that z ∈ (Z) is a solution of the equation ∆ a,c (z) = 0 i for any m ∈ Z, z p,m is a solution of the Chebyshev equation with parameter q(a, c; p); that is
As the boundedness of z is equivalent to the boundedness of the sequences z p,m , m = , . . . , p − , we can conclude that existence of bounded solutions for the equation ∆ a,c (z) = 0, depends only on the knowledge of the speci c value q(a, c; p). Since in [5, Theorem 3.3] the existence of this parameter was proved by induction the above result is not useful in practice. For this reason, most of the above mentioned paper was devoted to the explicit computation of the so-called Floquet function; that is, the function assigning the value q(a, c; p) to any a ∈ (Z; p) and c ∈ * (Z; p). Notice that, in fact, the value q(a, c; p) only depends on a(j), c(j), j = , . . . , p − . Once this function was obtained, the characterization of the existence of periodic solutions for the equation ∆ a,c (z) = 0 appears as a simple by-product, since from Lemma 2.1, they are characterized as being constant the sequences z p,m , ≤ m ≤ p − , see [5,Corollary 4.8]. So, the main novelty of this paper is to derive the characterization of the existence of bounded solutions for the equation ∆ a,c (z) = 0, from the value q(a, c; p). To do this, we need to introduce some notations and concepts. Given a binary multi-index of order p, α ∈ { , } p such that α = m ≥ , we consider ≤ i < ⋯ < i m ≤ p− such that α i = ⋯ = α i m = . Given p ∈ N * , we de ne the following subsets of the set { , } p of binary multiindexes of order p: In addition, if p ≥ , m = , . . . , ⌊ p ⌋ and α ∈ Λ m p , let ≤ i < ⋯ < i m ≤ p − be the indexes such that α i = ⋯ = α i m = . Then, we de ne the binary multi-indexᾱ of order p as where if i m = p − , thenᾱ p− =ᾱ = . Moreover, if α = ( , . . . , ); that is, if α ∈ Λ p , then we de nē α = ( , . . . , ). It is clear that, in any case, ᾱ = p − m.
We are now ready to show the expression for the value of q(a, c; p). In the sequel, we always assume that = and also the usual convention that empty sums and empty products are de ned as and , respectively.
Observe that when p = , the above identity becomes q(a, c; ) = a c ; that is, the value corresponding to the case in which the coe cients a and c are constant; or equivalent both have period p = .
Our main result appears now as a consequence of the Proposition 3.1 together with Lemma 2.1 and also the above Lemma. Theorem 4.3. Given p ∈ N * , a ∈ (Z; p) and c ∈ * (Z; p), then the equation   Clearly, the main di culty to apply the above characterizations is to obtain the binary multi-indexes involved in them. In general, this is a di cult task and, in fact, the number of multi-indexes in Λ j p , ≤ j ≤ ⌊ p ⌋, grows We end this paper with some speci c examples using the given characterization for the existence of bounded solutions for di erence equations with periodic coe cients with period up to . Remember that the case p = , the easiest case, has been analyzed in the previous sections. In particular, for the Mathieu which coe cient has period , the frequency is ω = m , where m ∈ Z is odd and hence ω = n + with n ∈ Z. Therefore, the coe cient is a(k) = E − λ cos(πk + θ), which implies that a( ) = E − λ cos(θ), whereas a( ) = E − λ cos(π + θ) = E + λ cos(θ). On the other hand, the frequency is ω = m where (m, ) = which implies that ω = n + r , where n ∈ Z and r = , . Therefore, the coe cient is given by a r (k) = E − λ cos π r k + θ , and hence a ( ) = a ( ) = E − λ cos(θ), . On the other hand, the frequency is ω = m where (m, ) = which implies that ω = n + r , where n ∈ Z and r = , . Therefore, the coe cient is given by a r (k) = E − λ cos π r k + θ , and hence