On the essential spectrum of an elliptic operator perturbed by a potential

Abstract

In [11] we made a study of the essential spectra of a strongly elliptic operator acting on L2(En). The object was to give sufficient conditions on a potential q to guarantee that the essential spectra of A + q are the same as those of A. The potential q was not assumed to have domain containing the domain of A, but merely to have a dense intersection with it (namely, the set C~ of infinitely differentiably functions with compact support). In this case one must consider some closed extension of A + q acting on C~o (there may be many). We found it convenient to employ regularly accretive extensions in the sense of Kato [13]. Sufficient conditions on q were given in order that a regularly accretive extension of A + q exist. Then further conditions were given in order that this extension have the same essential spectra as A. In this paper we are concerned with the same problem for A acting on LP(E~), 1 < p < oo. In this case strong ellipticity offers no advantage, so we drop this assumption. However, serious difficulties present themselves. On the one hand there is no known theory of closed extensions corresponding to Kato's regularly accretive extensions. On the other, the L 2 theory which is suitable for strongly elliptic operators no longer applies. The first difficulty is surmounted by developing a theory of extensions in an arbitrary Banach space which generalizes Kato's development. We call these operators "intermediate extensions". For a Hilbert space, regularly