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

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.

29 pages, corrections to section 3, added section 5

References in corpus (2)

• Spectrum of a non-self-adjoint operator associated with the periodic heat equation
• On the nature of ill-posedness of the forward-backward heat equation