Abstract
Abstract 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 coefficients, including the analysis of the so-called discrete Mathieu equations as particular cases.
Highlights
Discrete Schrödinger operators over nite or in nite paths have been subject of an intensive research over the last four decades
In addition, the coe cient a is given by a(k) = E − λ cos( πωk + θ), k ∈ Z, the operator Ha is called Mathieu operator and the parameters E, λ ∈ R, ω ∈ Q, θ ∈ [, π), are called the energy, coupling, frequency, and phase, respectively
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
Summary
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]. 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 extend the results to general second order linear di erence equations with periodic coe cients
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