Approximations of strongly continuous families of unbounded self-adjoint operators

TL;DR

This paper investigates how the spectra of strongly continuous families of unbounded self-adjoint operators can be approximated uniformly, especially under compactness conditions, with applications to plasma instability analysis.

Contribution

It introduces a method for uniformly approximating spectra of strongly continuous operator families under compactness assumptions.

Findings

Spectra may not vary continuously without additional assumptions.

Under compactness, spectra vary continuously with the family.

Constructed finite-dimensional symmetric approximations valid for the entire family.

Abstract

The problem of approximating the discrete spectra of families of self-adjoint operators that are merely strongly continuous is addressed. It is well-known that the spectrum need not vary continuously (as a set) under strong perturbations. However, it is shown that under an additional compactness assumption the spectrum does vary continuously, and a family of symmetric finite-dimensional approximations is constructed. An important feature of these approximations is that they are valid for the entire family uniformly. An application of this result to the study of plasma instabilities is illustrated.