# Reflectionless discrete Schr\"odinger operators are spectrally atypical

## TL;DR

This paper demonstrates that reflectionless discrete Schrödinger operators with nonconstant potentials have non-minimal shift dynamics and that only periodic operators can have zero, one, or two spectral gaps, highlighting their spectral atypicality.

## Contribution

It establishes new restrictions on the spectral properties and dynamics of reflectionless discrete Schrödinger operators, especially regarding their potential and spectral gaps.

## Key findings

- Operators with nonconstant potentials have non-minimal shift dynamics.
- Only periodic operators can have zero, one, or two spectral gaps.
- Certain spectral sets are not realizable as spectra of reflectionless operators.

## Abstract

We prove that, if an isospectral torus contains a discrete Schr\"odinger operator with nonconstant potential, the shift dynamics on that torus cannot be minimal. Consequently, we specify a generic sense in which finite unions of nondegenerate closed intervals having capacity one are not the spectrum of any reflectionless discrete Schr\"odinger operator. We also show that the only reflectionless discrete Schr\"odinger operators having zero, one, or two spectral gaps are periodic.