Self-adjoint Extensions of Restrictions

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 $S^{\ast }$ nor of the deficiency spaces of $S$ is required. Typically $A$ is a differential operator and $\mathbb{N}$ is the kernel of some trace (restriction) operator along a null subset. We parametrise the extensions by the bundle $π:\E(\fh)\to¶(\fh)$, where $¶(\fh)$ 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 $(Π,Θ)\in \E(\fh)$ 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 Schrödinger operators with point interactions and to elliptic boundary value problems are given.

Final version. To appear in Operators and Matrices

References in corpus (3)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Boundary triples and Weyl functions for singular perturbations of self-adjoint operators
• Self-Adjoint Extensions by Additive Perturbations

Cited by in corpus (5)

• Krein's Resolvent Formula for Self-Adjoint Extensions of Symmetric Second Order Elliptic Differential Operators
• Krein resolvent formulas for elliptic boundary problems in nonsmooth domains
• First order operators and boundary triples
• Spectral Analysis of a Two Body Problem with Zero Range Perturbation
• Graph-like models for thin waveguides with Robin boundary conditions