Scattering matrices and Weyl functions

Abstract

For a scattering system {AΘ, A0} consisting of self-adjoint extensions AΘ and A0 of a symmetric operator A with finite deficiency indices, the scattering matrix {SΘ(λ)} and a spectral shift function ξΘ are calculated in terms of the Weyl function associated with a boundary triplet for A*, and a simple proof of the Krein–Birman formula is given. The results are applied to singular Sturm–Liouville operators with scalar and matrix potentials, to Dirac operators and to Schrödinger operators with point interactions.