ar X iv : m at h - ph / 0 51 10 29 v 2 6 F eb 2 00 6 CONVERGENCE OF SCHRÖDINGER OPERATORS JOHANNES

We prove two limit relations between Schrödinger operators perturbed by measures. First, weak convergence of finite real-valued Radon measures µ n −→ m implies that the operators −∆ + ε 2 ∆ 2 + µ n in L 2 (R d , dx) converge to −∆ + ε 2 ∆ 2 + m in the norm resolvent sense, provided d ≤ 3. Second, for a large family, including the Kato class, of real-valued Radon measures m, the operators −∆ + ε 2 ∆ 2 + m tend to the operator −∆ + m in the norm resolvent sense as ε tends to zero. Explicit upper bounds for the rates of convergences are derived. Since one can choose point measures µ 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 Schrödinger operators. The approximation is illustrated by numerical calculations of eigenvalues for one simple example of measure m. In this paper we are going to analyze convergence of Schrödinger operators perturbed by measures. It is known that weak convergence of potentials implies norm-resolvent convergence of the corresponding one-dimensional Schrödinger operators. This result from [6] may be interesting for several reasons. For instance every finite real-valued Radon measure on R is the weak limit of a sequence of point measures with mass at only finitely many points. There exist efficient numerical methods for the 1