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 $\tau:D(A)\to\X$, X a Banach space, be a surjective linear map such that $\|\tau\phi\|_\X\le c \|A\phi\|_\H$. Supposing that \text{\rm Range}$ (\tau')\cap\H' =\{0\}$, we define a family $A^\tau_\Theta$ of self-adjoint operators which are extensions of the symmetric operator $A_{|\{\tau=0\}.}$. Any $\phi$ in the operator domain $D(A^\tau_\Theta)$ 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 $\|A\phi\|_\H$. These boundary conditions are written in terms of the map $\tau$, playing the role of a trace (restriction) operator, as $\tau\phireg=\Theta Q_\phi$, the extension parameter $\Theta$ being a self-adjoint operator from X' to X. The self-adjoint extension is then simply defined by $A^\tau_\Theta\phi:=A \phireg$. The case in which $A\phi=T*\phi$ is a convolution operator on LD, T a distribution with compact support, is studied in detail.