Rank one perturbations at infinite coupling (Q1891805)

The authors discuss rank one perturbations \(A_\alpha = A + \alpha (\varphi, \cdot) \varphi\), \(\alpha \in \mathbb{R}\), \(A \geq 0\) 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 \(L^2 (\mathbb{R}, d \rho_\infty)\). This generalizes results known for boundary condition dependence of Sturm-Liouville operators on half-lines to the abstract rank one case.