Correspondence of the eigenvalues of a non-self-adjoint operator to those of a self-adjoint operator

TL;DR

This paper establishes a correspondence between eigenvalues of a specific non-self-adjoint operator in fluid mechanics and those of a related self-adjoint operator, revealing their asymptotic distribution and enabling numerical computation.

Contribution

It demonstrates a novel spectral correspondence that allows analysis of non-self-adjoint operators via self-adjoint counterparts, advancing understanding in fluid mechanics applications.

Findings

Eigenvalues of the non-self-adjoint operator are real and accumulate at infinity.

The eigenvalue distribution asymptotically follows a specific pattern.

Numerical eigenvalues match previous calculations, validating the approach.

Abstract

We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at $\pm \infty$. We use this result to determine the asymptotic distribution of the eigenvalues and to compute some of the eigenvalues numerically. We compare these to earlier calculations by other authors.