On Symmetries in the Theory of Finite Rank Singular Perturbations

TL;DR

This paper develops a framework for analyzing symmetric singular perturbations of self-adjoint operators, with applications to Schrödinger operators and p-adic quantum models, highlighting the role of symmetries in such perturbations.

Contribution

It introduces a symmetry-based abstract framework for studying singular perturbations, connecting them to physically meaningful self-adjoint realizations.

Findings

Framework for symmetric singular perturbations of self-adjoint operators

Application to Schrödinger operators in L2(R^3)

Analysis of p-adic Schrödinger operators with point interactions

Abstract

For a nonnegative self-adjoint operator $A_0$ acting on a Hilbert space $\mathfrak{H}$ singular perturbations of the form $A_0+V, \ V=\sum_{1}^{n}{b}_{ij}<\psi_j,\cdot>\psi_i$ are studied under some additional requirements of symmetry imposed on the initial operator $A_0$ and the singular elements $\psi_j$. A concept of symmetry is defined by means of a one-parameter family of unitary operators $\sU$ that is motivated by results due to R. S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials $V$ and the corresponding self-adjoint realizations of $A_0+V$. The results are applied for the investigation of singular perturbations of the Schr\"{o}dinger operator in $L_2(\dR^3)$ and for the study of a (fractional) \textsf{p}-adic Schr\"{o}dinger type operator with point interactions.