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

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.

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

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.

An Indefinite Convection-Diffusion Operator With Real Spectrum

We confirm rigorously the conjecture, based on numerical and asymptotic evidence, that all the eigenvalues of a certain non-self-adjoint operator are real.