Singular Perturbations of Self-Adjoint Operators Associated with Rigged Hilbert Spaces

Abstract

Let A be an unbounded self-adjoint operator in a Hilbert separable space $H_0$ with rigging $H_ - \sqsupset H_0 \sqsupset H_ +$ such that $D(A) = H_ +$ in the graph norm (here, $D(A)$ is the domain of definition of A). Assume that $H_ +$ is decomposed into the orthogonal sum $H_ + = M \oplus N_ +$ so that the subspace $M_ +$ is dense in $H_0$. We construct and study a singularly perturbed operator A associated with a new rigging $H_ - \sqsupset H_0 \sqsupset \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} _ +$, where $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} _ + = M_ + = D(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A} )$, and establish the relationship between the operators A and A