Convergence of eigenvalues for a highly non-self-adjoint differential operator

TL;DR

This paper investigates how the eigenvalues of a family of non-self-adjoint differential operators, relevant in fluid mechanics, converge to a finite limit as a small parameter approaches zero, despite complex eigenvalue behavior for fixed parameters.

Contribution

It demonstrates the spectral convergence of a family of non-self-adjoint operators in fluid mechanics as the parameter tends to zero, revealing new asymptotic behavior.

Findings

Eigenvalues converge to N as epsilon approaches zero

Eigenvalue asymptotics are quadratic for fixed epsilon

Spectral analysis in non-self-adjoint operators

Abstract

In this paper we study a family of operators dependent on a small parameter $\epsilon > 0$, which arise in a problem in fluid mechanics. We show that the spectra of these operators converge to N as $\epsilon \to 0$, even though, for fixed $\epsilon > 0$, the eigenvalue asymptotics are quadratic.