Weak coupling limit for Schr\"odinger-type operators with degenerate kinetic energy for a large class of potentials

TL;DR

This paper advances the understanding of the weak coupling limit of eigenvalues in Schr"odinger-type operators with degenerate kinetic energy, employing the Tomas-Stein theorem to handle a broader class of potentials.

Contribution

It extends previous results by incorporating the Tomas-Stein theorem to analyze Schr"odinger operators with kinetic energy vanishing on submanifolds, broadening the class of potentials considered.

Findings

Improved eigenvalue asymptotics in the weak coupling limit.

Extended class of potentials with degenerate kinetic energy.

Application of harmonic analysis techniques to spectral problems.

Abstract

We improve results by Frank, Hainzl, Naboko, and Seiringer [12] and Hainzl and Seiringer [20] on the weak coupling limit of eigenvalues for Schr\"odinger-type operators whose kinetic energy vanishes on a codimension one submanifold. The main technical innovation that allows us to go beyond the potentials considered in [12, 20] is the use of the Tomas-Stein theorem.