Boundary Conditions for Singular Perturbations of Self-Adjoint Operators

2001/02/02 by Andrea Posilicano, Posilicano, Andrea
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Spectral Theory in Mathematical Physics #math-ph #math.FA #math.MP

paper · pdf · doi:10.48550/arxiv.math/0102018

Revised version. To appear in Operator Theory: Advances and Applications, vol. 132

openalex publication_date 2001/02/02 · arxiv created 2002/05/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A:D(A)⊆\H→\H be an injective self-adjoint operator and let τ:D(A)→\X, X a Banach space, be a surjective linear map such that ‖τϕ‖_\X≤ c ‖Aϕ‖_\H. Supposing that \rm Range (τ')∩\H' =\0\, we define a family A^τ_Θ of self-adjoint operators which are extensions of the symmetric operator A_|\τ=0\.. Any ϕ in the operator domain D(A^τ_Θ) is characterized by a sort of boundary conditions on its univocally defined regular component \phireg, which belongs to the completion of D(A) w.r.t. the norm ‖Aϕ‖_\H. These boundary conditions are written in terms of the map τ, playing the role of a trace (restriction) operator, as τ\phireg=ΘQ_ϕ, the extension parameter Θ being a self-adjoint operator from X' to X. The self-adjoint extension is then simply defined by A^τ_Θϕ:=A \phireg. The case in which Aϕ=T*ϕ is a convolution operator on LD, T a distribution with compact support, is studied in detail.

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