Self-adjoint Extensions of Restrictions

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dc.creatorPosilicano, Andrea
dc.date2007-03-27
dc.date2008-03-28
dc.date.accessioned2026-07-07T09:28:52Z
dc.date.available2026-07-07T09:28:52Z
dc.descriptionWe 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.
dc.descriptionFinal version. To appear in Operators and Matrices
dc.identifierhttps://arxiv.org/abs/math-ph/0703078
dc.identifierhttp://arxiv.org/abs/math-ph/0703078
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/157580
dc.subjectMathematical Physics
dc.subjectFunctional Analysis
dc.titleSelf-adjoint Extensions of Restrictions
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