Truncations of a class of pseudo-Hermitian operators

TL;DR

This paper studies non-Hermitian operators represented by infinite tridiagonal matrices, analyzing their eigenvalues and how well finite truncations approximate these eigenvalues, with implications for numerical simulations.

Contribution

It provides bounds on the convergence rate of eigenvalues of nonpositive type for truncated matrices of pseudo-Hermitian operators, enhancing numerical approximation reliability.

Findings

Eigenvalues are either real or complex conjugates.

Bounds for convergence rates of eigenvalues are established.

Numerical examples confirm theoretical results.

Abstract

We consider the class of non-Hermitian operators represented by infinite tridiagonal matrices, selfadjoint in an indefinite inner product space with one negative square. We approximate them with their finite truncations. Both infinite and truncated matrices have eigenvalues of nonpositive type: either a single one on the real axis or a couple of complex conjugate ones. As a tool to evaluate the reliability of the use of truncations in numerical simulations, we give bounds for the rate of convergence of their eigenvalues of nonpositive type. Numerical examples illustrate our results.