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.

Self-adjoint extensions of restricitons

POSILICANO, ANDREA
2008

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.
Self-adjoint extensions; Kre˘ın’s resolvent formula; elliptic boundary value problems
Posilicano, Andrea
File in questo prodotto:
File Dimensione Formato  
2008O&M.pdf

non disponibili

Tipologia: Documento in Post-print
Licenza: DRM non definito
Dimensione 290.01 kB
Formato Adobe PDF
290.01 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11383/10288
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? 71
social impact