We prove two limit relations between Schrodinger operators perturbed by measures. First, weak convergence of finite real-valued Radon measures $\mu_n \longrightarrow m$ implies that the operators $-\Delta + \varepsilon^2 \Delta^2 + \mu_n$ in $L^2(\mathbb{R}^d,dx)$ converge to $-\Delta + \varepsilon^2 \Delta^2 + m$ in the norm-resolvent sense, provided $d\le 3$. Second, for a large family, including the Kato class, of real-valued Radon measures m, the operators $-\Delta + \varepsilon^2 \Delta^2 + m$ tend to the operator $-\Delta +m$ in the norm-resolvent sense as $\varepsilon$ tends to zero. Explicit upper bounds for the rates of convergences are derived. Since one can choose point measures $\mu_n$ with mass at only finitely many points, a combination of both convergence results leads to an efficient method for the numerical computation of the eigenvalues in the discrete spectrum and corresponding eigenfunctions of Schrodinger operators.
['Johannes F. Brasche', 'Katerina Ozanova']
Convergence of Schrödinger Operators