On the Number of Closed Gaps of Discrete Periodic One-Dimensional Operators

TL;DR

This paper investigates the maximum number of closed spectral gaps in one-dimensional periodic discrete Schrödinger operators, providing bounds, exact computations for small periods, and characterizations of potentials with closed gaps.

Contribution

It establishes new bounds and exact values for the number of closed gaps in periodic Schrödinger operators, extending inverse spectral theory results.

Findings

Maximum one closed gap for periods four and five.

Maximum two closed gaps for period six.

Exact characterization of potentials with closed gaps.

Abstract

From the general inverse theory of periodic Jacobi matrices, it is known that a periodic Jacobi matrix of minimal period $p \geq 2$ may have at most $p-2$ closed spectral gaps. We discuss the maximal number of closed gaps for one-dimensional periodic discrete Schr\"odinger operators of period $p$. We prove nontrivial upper and lower bounds on this quantity for large $p$ and compute it exactly for $p \leq 6$. Among our results, we show that a discrete Schr\"odinger operator of period four or five may have at most a single closed gap, and we characterize exactly which potentials may exhibit a closed gap. For period six, we show that at most two gaps may close. In all cases in which the maximal number of closed gaps is computed, it is seen to be strictly smaller than $p-2$, the bound guaranteed by the inverse theory. We also discuss similar results for purely off-diagonal Jacobi matrices.