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 $S^*$ 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 $π:\E(\fh)\to¶(\fh)$, where $¶(\fh)$ denotes the set of orthogonal projections in the Hilbert space $\fh\simeq \D(A)/\N$ and $π^{-1}(Π)$ is the set of self-adjoint operators in the range of $Π$. The set of self-adjoint operators in $\fh$, i.e. $π^{-1}(1)$, parametrises the relatively prime extensions. Any $(Π,Θ)\in \E(\fh)$ determines a boundary condition in the domain of the corresponding extension $A_{Π,Θ}$ and explicitly appears in the formula for the resolvent $(-A_{Π,Θ}+z)^{-1}$. 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.