The Essential Spectrum of Elliptic Differential Operators in L p (R n )

Abstract

Introduction. The main aim of the present paper is to determine the essential spectrum and index of a class of elliptic differential operators in L'(Rn) (see Definition 1.3). By a well-known theorem of perturbation theory, the essential spectrum and index of an operator A are unchanged under addition of an operator B, which is compact relative to A (see Definition 1.1). ?1 contains without proof the definitions and main theorems of this theory. In ?3 we consider the case of an elliptic differential operator Pp with constant coefficients in LP(Rn) perturbed by lower-order terms. The spectrum of the constant coefficient operator is given in Theorem 3.5. Then we determine a rather large class of operators, which are compact with respect to Pp. Theorem 3.9 contains the main result on the essential spectrum and index of the perturbed constant coefficient operator. The preliminary work leading to the compactness conditions is done in ?2. The graph norm of the elliptic constant coefficient operator is equivalent to the Wy-norm (Definition 2.2). Therefore, the problem of finding compactness conditions is essentially reduced to the problem of finding conditions in order that the embedding of W (Rn) in LP(Rn, b) with a weight function b be compact. The main result in this direction is stated in Lemmas 2.11 and 2.15.