Andrea Posilicano
Self-Adjoint Extensions by Additive Perturbations
(43K, ams-latex)

ABSTRACT.  Let $A_N$ be the restriction of $A$ to $N$, where 
$A:D(A)\subseteq H\to H$ is a 
self-adjoint 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 self-adjoint 
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 self-adjoint 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 self-adjoint extensions is given.