On Minimal Entire Solutions of the One-Dimensional Difference Schrödinger Equation with the Potential υ(z) = e−2πiz

Abstract

Let z ∈ ℂ be a complex variable, and let h ∈ (0, 1) and p ∈ ℂ be parameters. For the equation ψ(z + h) + ψ(z − h) + e^−2πizψ(z) = 2 cos(2πp)ψ(z), solutions having the minimal possible growth simultaneously as Im z → ∞ and as Im z →  − ∞ are studied. In particular, it is shown that they satisfy one more difference equation ψ(z + 1) + ψ(z − 1) + e^−2πiz/hψ(z) = 2 cos(2πp/h)ψ(z).