Isospectral Mathieu-Hill Operators

Abstract

In this paper we prove that the spectrum of the Mathieu-Hill Operators with potentials ae^{-i2πx}+be^{i2πx} and ce^{-i2πx}+de^{i2πx} 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.

References in corpus (2)

• Spectral Analysis of the Non-self-adjoint Mathieu-Hill Operator
• Improved asymptotics of the spectral gap for the Mathieu operator

Cited by in corpus (6)

• On the spectral properties of the Schrodinger operator with a periodic PT-symmetric potential
• The Spectrum of the Hamiltonian with a PT-symmetric Periodic Optical Potential
• Spectral Analysis of the Schrodinger Operator with an Optical Potential
• On the Simplicity of the Eigenvalues of the Non-self-adjoint Mathieu-Hill Operators
• Spectral Analysis of the Non-self-adjoint Mathieu-Hill Operator
• Asymptotic formulas for spectral gaps and deviations of Hill and 1D Dirac operators