Andrea Posilicano
Boundary 
Conditions for Singular Perturbations of Self-Adjoint Operators
(37K, amslatex)

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.