Self-Adjoint Extensions by Additive Perturbations

TL;DR

This paper develops a framework for understanding self-adjoint extensions of symmetric operators through additive perturbations, explicitly connecting boundary conditions, Krein's formula, and von Neumann's theory.

Contribution

It introduces a decomposition of self-adjoint extensions into a sum of a closed extension and an explicit boundary-dependent operator, linking classical theories in a new way.

Findings

Decomposition of self-adjoint extensions into additive components

Explicit boundary condition representation of extensions

Connection with Krein's resolvent formula and von Neumann's theory

Abstract

Let $A_\N$ be the symmetric operator given by the restriction of $A$ to $\N$, where $A$ is a self-adjoint operator on the Hilbert space $\H$ and $\N$ is a linear dense set which is closed with respect to the graph norm on $D(A)$, the operator domain of $A$. We show that any self-adjoint extension $A_\Theta$ of $A_\N$ such that $D(A_\Theta)\cap D(A)=\N$ can be additively decomposed by the sum $A_\Theta=\A+T_\Theta$, where both the operators $\A$ and $T_\Theta$ take values in the strong dual of $D(A)$. The operator $\A$ is the closed extension of $A$ to the whole $\H$ whereas $T_\Theta$ is explicitly written in terms of a (abstract) boundary condition depending on $\N$ and on the extension parameter $\Theta$, a self-adjoint operator on an auxiliary Hilbert space isomorphic (as a set) to the deficiency spaces of $A_\N$. The explicit connection with both Kre\u\i n's resolvent formula and von Neumann's theory of self-adjoint extensions is given.