Cwikel-Lieb-Rozenblum type estimates for the Pauli and magnetic Schr\"odinger operator in dimension two

TL;DR

This paper establishes sharp bounds on the number of negative eigenvalues for two-dimensional Pauli and magnetic Schr"odinger operators, considering effects of spin-orbit coupling, advancing spectral theory in quantum physics.

Contribution

It extends Cwikel-Lieb-Rozenblum inequalities to two-dimensional Pauli operators, including spin-orbit effects, with sharp bounds in various coupling regimes.

Findings

Derived sharp upper bounds for negative eigenvalues in 2D Pauli operators.

Established different bounds depending on spin-orbit coupling presence.

Provided comprehensive estimates for magnetic Schr"odinger operators.

Abstract

We prove a Cwikel-Lieb-Rozenblum type inequality for the number of negative eigenvalues of Pauli operators in dimension two. The resulting upper bound is sharp both in the weak as well as in the strong coupling limit. We also derive different upper bounds for magnetic Schr\"odinger operators. The nature of the two estimates depends on whether or not the spin-orbit coupling is taken into account.