Singularly Perturbed Self-Adjoint Operators in Scales of Hilbert spaces

TL;DR

This paper investigates finite rank perturbations of semi-bounded self-adjoint operators within Hilbert space scales, introducing a quasi-boundary value space concept to describe various operator realizations, with applications to Schrödinger operators with zero-range potentials.

Contribution

It introduces the quasi-boundary value space framework for describing self-adjoint realizations of perturbations in Hilbert space scales.

Findings

Describes self-adjoint operator realizations via a unified formula.

Analyzes the Schrödinger operator with zero-range potential in Sobolev spaces.

Provides a new approach to singular perturbations in spectral theory.

Abstract

Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and singular perturbations of A by the same formula. As an application the one-dimensional Schr\"{o}dinger operator with generalized zero-range potential is considered in the Sobolev space W^p_2(\mathbb{R}), p\in\mathbb{N}.