Whittaker-Hill equation and semifinite-gap Schroedinger operators

TL;DR

This paper constructs explicit semifinite-gap Schrödinger operators using Darboux transformations on the Whittaker-Hill equation, analyzing their spectral properties and potential regularity.

Contribution

It introduces a method to generate semifinite-gap operators via Darboux transformations, providing explicit examples and spectral analysis.

Findings

Explicit semifinite-gap operators constructed

Criteria for potential regularity established

Spectral properties of new operators analyzed

Abstract

A periodic one-dimensional Schroedinger operator is called semifinite-gap if every second gap in its spectrum is eventually closed. We construct explicit examples of semifinite-gap Schroedinger operators in trigonometric functions by applying Darboux transformations to the Whittaker-Hill equation. We give a criterion of the regularity of the corresponding potentials and investigate the spectral properties of the new operators.