A family of Schr\"odinger operators whose spectrum is an interval

TL;DR

This paper demonstrates that certain Schrödinger operators with quasi-periodic potentials have spectra forming continuous intervals, expanding understanding of spectral properties in quantum mechanics.

Contribution

The paper introduces a method to show that the spectrum of Schrödinger operators with specific quasi-periodic potentials is an interval, using approximation techniques.

Findings

Spectrum of the operator is an interval for the given potential.

Applicable to potentials with irrational frequency parameters.

Provides a new approach to spectral analysis of quasi-periodic operators.

Abstract

By approximation, I show that the spectrum of the Schr\"odinger operator with potential $V(n) = f(n\rho \pmod 1)$ for f continuous and $\rho > 0$, $\rho \notin \N$ is an interval.