Boundary Conditions for Singular Perturbations of Self-adjoint Operators

Abstract

Let $A:D(A) \subseteq \mathcal{H} \to \mathcal{H}$ be an injective self-adjoint operator and let $\tau :\;D(A) \to \chi$ ,χ a Banach space, be a surjective linear map such that ${\left\| {\tau \phi } \right\|_\chi } \leqslant c{\left\| {A\phi } \right\|_H}$ Supposing that Kernel T is dense in $\mathcal{H}$ we define a family $A_\Theta ^\tau$ of self-adjoint operators which are extensions of the symmetric operator ${A_{\left| {\left\{ {\tau = 0} \right\}} \right.}}$ Any Ø in the operator domain $D(A_\Theta ^\tau )$ is characterized by a sort of boundary conditions on its univocally defined regular component Ø_reg, which belongs to the completion of D(A) w.r.t. the norm ${\left\| {A\phi } \right\|_\mathcal{H}}$ These boundary conditions are written in terms of the map T, playing the role of a trace (restriction) operator, as $\tau {\phi _{reg}} = \Theta {Q_\phi }$ the extension parameter >Θ being a self-adjoint operator from χ’ to χ. The self-adjoint extension is then simply defined by $A_\Theta ^\tau \phi : = {\rm A}{\phi _{reg}}$ The case in which $A\phi = \Psi *\phi$ is a convolution operator on $L^2 (\mathbb{R}^n )$ ,Ψ a distribution with compact support, is studied in detail.