Convergence of Schrödinger Operators

For a large class, containing the Kato class, of real-valued Radon measures m on R the operators −∆ + ε∆ + m in L(R, dx) tend to the operator −∆ +m in the norm resolvent sense, as ε tends to zero. If d ≤ 3 and a sequence (μn) of finite real-valued Radon measures on R converges to the finite real-valued Radon measure m weakly and, in addition, supn∈N μ ± n (R) < ∞, then the operators −∆ + ε∆ + μn converge to −∆ + ε∆ + m in the norm resolvent sense. Explicit upper bounds for the rates of convergences are derived. One can choose point measures μn with mass at only finitely many points so that 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. This article has been submitted to Journal of Mathematical Physics.