© 1998 by London Mathematical Society
© The London Mathematical Society
Kato Class Potentials for Higher Order Elliptic Operators
Department of Mathematics, King's College London Strand, London WC2R 2LS. E-mail: E.Brian.Davies{at}kcl.ac.uk
Mathematisches Institut, Universität München Theresienstraße 39, D-80333 München, Germany. E-mail: hinz{at}rz.mathematik.uni-muenchen.de
Received 15 April 1996.
Our goal in this paper is to determine conditions on a potential V which ensure that an operator such as
H:=(–
)m+V (1)
acting on L2(RN) defines a semigroup in Lp(RN) for various values of p including p=1. The operator is defined as a quadratic form sum. That is, we put
 |
for 
(all integrals are on RN 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 Wm,2. We then assume that the potential has quadratic form bound less than 1 with respect to Q0, and define
 |
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
0 is extendable to a strongly continuous one-parameter semigroup on Lp for other values of p, and if so whether one can describe the domain and spectrum of its generator.