Gesztesy F., Simon B.
Rank One Perturbations at Infinite Coupling
(17K, AMSTeX)
ABSTRACT. We discuss rank one perturbations $A_{\alpha}=A+\alpha
(\varphi, \cdot)\varphi, \alpha\in\Bbb R, A\geq 0$ self-adjoint. Let
$d\mu_{\alpha}(x)$ be the spectral measure defined by $(\varphi,
(A_{\alpha}-z)^{-1}\varphi)=\int\, d\mu_{\alpha}(x)/(x-z)$. We prove there
is a measure $d\rho_{\infty}$ which is the weak limit of
$(1+\alpha^{2})\,d\mu_{\alpha}(x)$ as $\alpha\to\infty$. If $\varphi$
is cyclic for $A$, then $A_\infty$, the strong resolvent limit of
$A_\alpha$, is unitarily equivalent to multiplication by $x$ on
$L^{2}(\Bbb R, d\rho_{\infty})$. This generalizes results known for
boundary condition dependence of Sturm-Liouville operators on half-lines
to the abstract rank one case.