01-52 Andrea Posilicano

Boundary Conditions for Singular Perturbations of Self-Adjoint Operators (37K, amslatex) Feb 2, 01

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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 Range}$(\tau')\cap\H' =\left\{0\right\}$, we define a family $A^\tau_\Theta$ of self-adjoint operators which are extensions of the symmetric operator $A_{\left|\{\tau=0\}\right.}$. 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 $L^2$, $T$ a distribution with compact support, is studied in detail.

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