The rotation number for almost periodic potentials with jump discontinuities and $\delta$-interactions

TL;DR

This paper introduces a rotation number concept for one-dimensional Schrödinger operators with generalized almost periodic potentials, including jump discontinuities and delta interactions, extending the theory of almost periodicity.

Contribution

It develops a new rotation number framework for Schrödinger operators with complex potentials, broadening the scope of spectral analysis in such systems.

Findings

Defined a rotation number for these operators

Extended almost periodicity to potentials with discontinuities

Provided a foundation for spectral analysis of complex potentials

Abstract

We consider one-dimensional Schr\"odinger operators with generalized almost periodic potentials with jump discontinuities and $\delta$-interactions. For operators of this kind we introduce a rotation number in the spirit of Johnson and Moser. To do this, we introduce the concept of almost periodicity at a rather general level, and then the almost periodic function with jump discontinuities and $\delta$-interactions as an application.