Self-adjoint Extensions of Restrictions

2007/03/27 by Andrea Posilicano, Posilicano, Andrea · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics #math-ph #math.FA #math.MP

paper · pdf · doi:10.48550/arxiv.math-ph/0703078

Final version. To appear in Operators and Matrices

openalex publication_date 2007/03/27 · arxiv created 2008/03/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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)⊆\H→\H to the dense, closed with respect to the graph norm, subspace \N⊂ \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)→¶(\fh), where ¶(\fh) denotes the set of orthogonal projections in the Hilbert space \fh≃ \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)∈ \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.

Citations

• Boundary triples and Weyl functions for singular perturbations of self-adjoint operators
• Self-Adjoint Extensions by Additive Perturbations

Cited by

• Generalized boundary triples for adjoint pairs with applications to non-self-adjoint Schrödinger operators
• Spectral analysis of a two body problem with zero-range perturbation

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