M operators: a generalisation of Weyl–Titchmarsh theory

Abstract

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 −Δ+ 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≡ 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.