Approximation by point potentials in a magnetic field

TL;DR

This paper demonstrates that magnetic Schrödinger operators with measure perturbations can be approximated by operators with point potentials, providing a new method for spectral analysis and numerical calculations.

Contribution

It introduces an explicit Krein-like formula for resolvents and shows strong resolvent convergence to simplify spectral computations.

Findings

Operators with measure perturbations can be approximated by point potential operators.

The spectral problem becomes solvable for the approximating operators.

Numerical calculations illustrate the approach for potentials supported on a circle.

Abstract

We discuss magnetic Schrodinger operators perturbed by measures from the generalized Kato class. Using an explicit Krein-like formula for their resolvent, we prove that these operators can be approximated in the strong resolvent sense by magnetic Schrodinger operators with point potentials. Since the spectral problem of the latter operators is solvable, one in fact gets an alternative way to calculate discrete spectra; we illustrate it by numerical calculations in the case when the potential is supported by a circle.