KATO CLASS POTENTIALS FOR HIGHER ORDER ELLIPTIC OPERATORS

Abstract

Our goal in this paper is to determine conditions on a potential V which ensure that an operator such as

formula here

acting on L²(R^N) defines a semigroup in L^p(R^N) for various values of p including p=1. The operator is defined as a quadratic form sum. That is, we put

formula here

for f∈C^∞_c (all integrals are on R^N and are with respect to Lebesgue measure), and note that the closure of the form is non-negative and has domain equal to the Sobolev space W^m,2. We then assume that the potential has quadratic form bound less than 1 with respect to Q₀, and define

formula here

This form is closed and is associated with a semibounded self-adjoint operator H in L² (see [17, p. 348; 5, Theorem 4.23]). One can then ask whether the semigroup e^−Ht defined on L² for t[ges ]0 is extendable to a strongly continuous one-parameter semigroup on L^p for other values of p, and if so whether one can describe the domain and spectrum of its generator.