Numerical Linear Algebra and Solvability of Partial Differential Equations

Abstract

 It was observed long ago that the obstruction to the accurate computation of eigenvalues of large non-self-adjoint matrices is inherent in the problem. The basic idea is that the resolvent of a highly non-normal operator can be very large far away from the spectrum. This leads to an easily observable fact that algorithms for locating eigenvalues will typically find some ``false eigenvalues''. These false eigenvalues also explain one of the most surprising phenomena in linear PDEs, namely the fact (discovered by Hans Lewy in 1957, in Berkeley) that one cannot always locally solve the PDE P u = f. Almost immediately after that discovery, Hörmander provided an explanation of Lewy's example showing that almost all operators with non-constanct complex valued coefficients are not locally solvable. In modern language, that was done by considering the essentially dual problem of existence of non-propagating singularities. The purpose of this article is to review this work in the context of ``almost eigenvalues'' and from the point of view of semi-classical analysis.