On Schrödinger operators perturbed by fractal potentials (Q1580726)

Let \(\Gamma\subset \mathbb{R}^n\) be a self-similar fractal with positive \(W^1_2\)-capacity and let \(V: W^1_2(\mathbb{R}^n)\to W^{-1}_2(\mathbb{R}^n)\) be supported on \(\Gamma\). The paper discusses the problem of defining a self-adjoint realization in \(L^2(\mathbb{R}^n)\) of the expression \(-\Delta_{\beta, V,\Gamma}= -\Delta+\beta V\), \(\beta\in \mathbb{R}\). The operator defined is lower semi-bounded, has essential spectrum \([0,\infty)\) and conditions are given which ensure that \(-1\) is its lowest eigenvalue. Furthermore, it is proved that as \(\beta\to \pm\infty\), \(-\Delta_{\beta,V,\Gamma}\) converges in the strong resolvent sense to the Friedrichs extension of \(-\Delta\) on \(\{f\in W^1_2(\mathbb{R}^n): f=0\) on \(\Gamma\}\).