 0152 Andrea Posilicano
 Boundary
Conditions for Singular Perturbations of SelfAdjoint Operators
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Feb 2, 01

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Abstract. Let $A:D(A)\subseteq\H\to\H$ be an injective
selfadjoint 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 selfadjoint 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 selfadjoint operator
from $\X'$ to $\X$. The selfadjoint 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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