Boundary Conditions for Singular Perturbations of Self-Adjoint Operators

Abstract

Let $A:D(A)\subseteq\H\to\H$ be an injective self-adjoint operator and let $τ:D(A)\to\X$, X a Banach space, be a surjective linear map such that $\|τϕ\|_\X\le c \|Aϕ\|_\H$. Supposing that \text{\rm Range}$ (τ')\cap\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.