Isospectral Mathieu-Hill Operators

TL;DR

This paper establishes a condition under which two Mathieu-Hill operators have identical spectra, and derives explicit formulas for eigenvalues and eigenfunctions for specific potentials, advancing spectral theory understanding.

Contribution

It proves a spectral equivalence condition for Mathieu-Hill operators with certain potentials and provides explicit eigenvalue and eigenfunction formulas for Gasymov's potential.

Findings

Spectral equivalence if and only if ab=cd for potentials

Explicit eigenvalues and eigenfunctions for Gasymov's potential

Extension of Harrell-Avron-Simon formula

Abstract

In this paper we prove that the spectrum of the Mathieu-Hill Operators with potentials ae^{-i2{\pi}x}+be^{i2{\pi}x} and ce^{-i2{\pi}x}+de^{i2{\pi}x} are the same if and only if ab=cd, where a,b,c and d are complex numbers. This result implies some corollaries about the extension of Harrell-Avron-Simon formula. Moreover, we find explicit formulas for the eigenvalues and eigenfunctions of the t-periodic boundary value problem for the Hill operator with Gasymov's potential.