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Scattering matrices and Weyl functions

For a scattering system $\{A_Θ,A_0\}$ consisting of selfadjoint extensions $A_Θ$ and $A_0$ of a symmetric operator $A$ with finite deficiency indices, the scattering matrix $\{S_\gT(\gl)\}$ and a spectral shift function $ξ_Θ$ are calculated in terms of the Weyl function associated with the 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.