Asymptotic Analysis of Non-self-adjoint Hill Operators

TL;DR

This paper derives uniform asymptotic formulas for eigenvalues and eigenfunctions of non-self-adjoint Hill operators with periodic potentials, establishing conditions for finiteness of spectral singularities and asymptotic spectrality.

Contribution

It provides new uniform asymptotic formulas for eigenvalues/eigenfunctions and identifies conditions ensuring finite spectral singularities and asymptotic spectrality of Hill operators.

Findings

Eigenvalues and eigenfunctions asymptotics derived

Spectral singularities are finite under certain conditions

Operator is asymptotically spectral if conditions are met

Abstract

We obtain the uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L_{t}(q) with a potential q\inL_{1}[0,1] and with t-periodic boundary conditions, t\in(-{\pi},{\pi}]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L_{2}(-\infty,\infty) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies the sufficient conditions.