Convergence of Hill's method for nonselfadjoint operators

TL;DR

This paper proves the convergence of Hill's method for approximating spectra of nonselfadjoint periodic differential operators using a generalized Evans function and Fredholm determinants, extending previous results to a broader class of operators.

Contribution

It introduces a new proof of convergence for Hill's method applicable to nonselfadjoint operators with positive definite principal coefficients, including cases previously unaddressed.

Findings

Convergence in location and multiplicity of spectra is established.

The method applies to a large class of nonselfadjoint operators.

The approach involves a generalized Evans function and Fredholm determinants.

Abstract

By the introduction of a generalized Evans function defined by an appropriate 2-modified Fredholm determinant, we give a simple proof of convergence in location and multiplicity of Hill's method for numerical approximation of spectra of periodic-coefficient ordinary differential operators. Our results apply to operators of nondegenerate type, under the condition that the principal coefficient matrix be symmetric positive definite (automatically satisfied in the scalar case). Notably, this includes a large class of nonselfadjoint operators, which were previously not treated. The case of general coefficients depends on an interesting operator-theoretic question regarding properties of Toeplitz matrices.