M operators:: a generalisation of Weyl-Titchmarsh theory

The theory of the Weyl-Titchmarsh m function for second-order ordinary differential operators is generalized and applied to partial differential operators of the form -Delta + q(x) acting in three space dimensions. Weyl operators M(z) are defined as maps from L-2(S-1) to L-2(S-1) (S-1 equivalent to unit sphere in R-3) for exterior and interior boundary value problems, and for the partial differential operator acting in L-2(R-3), with the standard Weyl-Titchmarsh m function recovered in the special case that q is spherically symmetric. The analysis is carried out rather explicitly, allowing for the determination of precise norm bounds for M operators and for the proof of higher dimensional analogues of a number of the fundamental results of standard Weyl-Titchmarsh theory. (C) 2004 Elsevier B.V. All rights reserved.