Self-adjoint Extensions of Restrictions

TL;DR

This paper introduces a straightforward method to characterize all self-adjoint extensions of a symmetric operator restricted to a subspace, providing explicit resolvent formulas and connecting with existing theories, with applications in quantum graphs and boundary problems.

Contribution

It offers a new parametrization of self-adjoint extensions that does not require deficiency space knowledge, simplifying the extension theory and linking it to boundary conditions.

Findings

Explicit parametrization of all self-adjoint extensions.

Derivation of resolvent formulas for the extensions.

Applications to quantum graphs and boundary value problems.

Abstract

We provide a simple recipe for obtaining all self-adjoint extensions, together with their resolvent, of the symmetric operator $S$ obtained by restricting the self-adjoint operator $A:\D(A)\subseteq\H\to\H$ to the dense, closed with respect to the graph norm, subspace $\N\subset \D(A)$. Neither the knowledge of $S^*$ nor of the deficiency spaces of $S$ is required. Typically $A$ is a differential operator and $\N$ is the kernel of some trace (restriction) operator along a null subset. We parametrise the extensions by the bundle $\pi:\E(\fh)\to\P(\fh)$, where $\P(\fh)$ denotes the set of orthogonal projections in the Hilbert space $\fh\simeq \D(A)/\N$ and $\pi^{-1}(\Pi)$ is the set of self-adjoint operators in the range of $\Pi$. The set of self-adjoint operators in $\fh$, i.e. $\pi^{-1}(1)$, parametrises the relatively prime extensions. Any $(\Pi,\Theta)\in \E(\fh)$ determines a boundary condition in the domain of the corresponding extension $A_{\Pi,\Theta}$ and explicitly appears in the formula for the resolvent $(-A_{\Pi,\Theta}+z)^{-1}$. The connection with both von Neumann's and Boundary Triples theories of self-adjoint extensions is explained. Some examples related to quantum graphs, to Schr\"odinger operators with point interactions and to elliptic boundary value problems are given.