01-154 Andrea Posilicano

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

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