Relative stability of singular spectrum

Abstract

There are classical theorems of analysis which, given certain conditions on a perturbation, assert stability of the essential and absolutely continuous components of the spectrum of a self-adjoint operator. Whereas the singular component is known to be highly volatile under the weakest of all possible perturbations, -- rank one. In this note I announce a theorem which asserts that, nevertheless, the singular component of spectrum in an open interval is in a sense \emph{relatively stable} provided the limiting absorption principle (LAP) holds in the interval. One of the benefits of this result is the provision of a method for disproving LAP where it is suspected to fail.