Sub-criticality of Schroedinger Systems with Antisymmetric Potentials

TL;DR

This paper demonstrates that Schrödinger systems with antisymmetric potentials in higher dimensions can be expressed in divergence form, leading to improved regularity results for solutions in certain Lebesgue spaces.

Contribution

It proves the sub-criticality of Schrödinger systems with antisymmetric potentials in dimensions m ≥ 3, showing solutions gain higher regularity.

Findings

Solutions in L^{m/(m-2)} are in W^{2,q}_{loc} for any q<m/2

Schrödinger systems with antisymmetric potentials can be written in divergence form

Regularity results extend to higher-dimensional systems with specific antisymmetry conditions

Abstract

Let $m$ be an integer larger or equal to 3. We prove that Schroedinger systems on $B^m$ with $L^{m/2}-$antisymmetric potential $\Omega$ of the form $$ -\Delta v=\Omega v $$ can be written in divergence form and we deduce that solutions $v$ in $L^{m/(m-2)}$ are in fact $W^{2,q}_{loc}$ for any $q<m/2$.