Perturbation of continuous spectra and unitary equivalence

Abstract

This chapter is concerned with the perturbation theory for continuous spectra. The operators considered are mostly selfadjoint. The stability of the continuous spectrum under a small perturbation has been studied rather extensively, though the results are by no means satisfactory. It is known that the continuous spectrum is rather unstable, even under degenerate perturbations. In this respect it is much worse-behaved than the essential spectrum (which is in general larger than the continuous spectrum). On the other hand, the absolutely continuous spectrum (which is in general smaller than the continuous spectrum) is stable under certain restricted perturbations; furthermore, the absolutely continuous parts of the perturbed and the unperturbed operators are seen to be unitarily equivalent.