Some Remarks on 1D Schrödinger Operators With Localized Magnetic and Electric Potentials

Abstract

One-dimensional Schrödinger operators with singular perturbed magnetic and electric potentials are considered. We study the strong resolvent convergence of two families of the operators with potentials shrinking to a point. Localized δ-like magnetic fields are combined with δ ′ -like perturbations of the electric potentials as well as localized rank-two perturbations. The limit results obtained heavily depend on zero-energy resonances of the electric potentials. In particular, the approximation for a wide class of point interactions in one dimension is obtained.

The physical interpretation of point interactions in one-dimensional relativistic quantum mechanics

“…There have been claims in the literature that the phase φ and, consequently, also the magnetic potential parameter A 1 (given (58) and, in the non-relativistic case, [29]), are trivial in the stationary case [30]. Although, for the Schrödinger theory Golovaty [25] has recently shown, using a norm-convergent regularization, an example to the contrary-including the fact that the magnetic potential can affect all parameters in the b.c. Λ-matrix.…”

“…There are, however, mathematically rigorous approaches to the regularization of PI addressing these issues, e.g. [22][23][24][25][26].…”