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 −Δ+q(x) acting in three space dimensions. Weyl operators M(z) are defined as maps from L2(S1) to L2(S1) (S1≡ unit sphere in R3) for exterior and interior boundary value problems, and for the partial differential operator acting in L2(R3), 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.