Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Scattering Matrices for Close Singular Selfadjoint Perturbations of Unbounded Selfadjoint Operators (2203.13163v1)

Published 24 Mar 2022 in math.SP

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.

Summary

We haven't generated a summary for this paper yet.