 01154 Andrea Posilicano
 SelfAdjoint Extensions by Additive Perturbations
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Apr 24, 01

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Abstract. Let $A_N$ be the restriction of $A$ to $N$, where
$A:D(A)\subseteq H\to H$ is a
selfadjoint operator and $N\subsetneq D(A)$ is a linear dense set which is
closed with respect to the graph
norm on $D(A)$. We show how to define any relatively prime selfadjoint
extension $A_\Theta:D(\Theta)\subseteq H\to H$ of $A_N$ by
$A_\Theta:=\A+T_\Theta$,
where both the
operators $\A$ and $T_\Theta$ take values in the strong dual of
$D(A)$. The operator $\A$ is the closed extension of $A$ to the whole
$\H$ whereas $T_\Theta$
is explicitly written in terms of a (abstract) boundary condition
depending on $N$ and on the extension parameter $\Theta$, a selfadjoint operator on
an auxiliary Hilbert space isomorphic (as a set) to the
deficiency spaces of $A_N$.
The explicit connection with both Krein's resolvent formula and von
Neumann's theory of selfadjoint extensions is given.
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