Eigenvalues of non-selfadjoint operators: A comparison of two approaches

TL;DR

This paper compares two mathematical approaches—complex analysis and operator theory—for analyzing eigenvalue distributions of non-selfadjoint operators, with applications to Jacobi and Schrödinger operators.

Contribution

It provides a detailed comparison of two methods for studying eigenvalues of non-selfadjoint operators, expanding understanding of their spectral properties.

Findings

Both methods yield general eigenvalue distribution results.

Applications to non-selfadjoint Jacobi and Schrödinger operators demonstrate practical relevance.

Discussion of future research directions in spectral theory.

Abstract

The central problem we consider is the distribution of eigenvalues of closed linear operators which are not selfadjoint, with a focus on those operators which are obtained as perturbations of selfadjoint linear operators. Two methods are explained and elaborated. One approach uses complex analysis to study a holomorphic function whose zeros can be identified with the eigenvalues of the linear operator. The second method is an operator theoretic approach involving the numerical range. General results obtained by the two methods are derived and compared. Applications to non-selfadjoint Jacobi and Schr\"odinger operators are considered. Some possible directions for future research are discussed.