Scattering Matrices for Close Singular Selfadjoint Perturbations of Unbounded Selfadjoint Operators

Abstract: In this paper, we consider an unbounded selfadjoint operator $A$ and its selfadjoint perturbations in the same Hilbert space $\mathcal{H}$. As S.Albeverio and P. Kurosov (2000), we call a selfadjoint operator $A_{1}$ the singular perturbation of $A$ if $A_{1}$ and {A} have different domains $\mathcal{D}(A),\mathcal{D}(A_{1})$ but $A=A_{1}$ on $\mathcal{D}(A)\cap\mathcal{D}(A_{1})$. Assuming that $A$ has absolutely continuous spectrum and the difference of resolvents $R_{z}(A_{1}) -R_{z}(A)$ of $A_{1}$ and $A$ for non-real $z$ is a trace class operator we find the explicit expression for the scattering matrix for the pair $A, A_{1}$ through the constituent elements of the Krein formula for the resolvents of this pair. As an illustration, we find the scattering matrix for the standardly defined Laplace operator in $L_{2}\left(\mathbf{R}_{3}\right)$ and its singular perturbation in the form of an infinite sum of zero-range potentials.