Boundary triples and Weyl functions for singular perturbations of self-adjoint operators

TL;DR

This paper develops a framework using boundary triples and Weyl functions to analyze singular perturbations of self-adjoint operators, enabling explicit descriptions of their extensions and resolvents.

Contribution

It introduces a method to construct boundary triples and Weyl functions for singular perturbations of self-adjoint operators, facilitating the study of their extensions.

Findings

Constructed boundary triples for the adjoint operator

Derived explicit formulas for Weyl functions

Provided a way to describe self-adjoint extensions and resolvents

Abstract

Given the symmetric operator $A_N$ obtained by restricting the self-adjoint operator $A$ to $N$, a linear dense set, closed with respect to the graph norm, we determine a convenient boundary triple for the adjoint $A_N^*$ and the corresponding Weyl function. These objects provide us with the self-adjoint extensions of $A_N$ and their resolvents.