Counterexamples to Strichartz estimates for the magnetic Schroedinger equation

TL;DR

This paper constructs explicit magnetic potentials with slow decay that violate Strichartz estimates for the Schrödinger equation in dimensions three and higher, challenging previous assumptions about solution behavior.

Contribution

It provides explicit counterexamples of magnetic potentials with less than Coulomb decay where Strichartz estimates fail for all admissible solutions.

Findings

Counterexamples with slow decay potentials

Failure of Strichartz estimates in all admissible ranges

Implications for magnetic Schrödinger equation analysis

Abstract

In space dimension $n\geq3$, we consider the magnetic Schr\"odinger Hamiltonian $H=-(\nabla-iA(x))^2$ and the corresponding Schr\"odinger equation i\partial_tu+Hu=0. We show some explicit examples of potentials $A$, with less than Coulomb decay, for which any solution of this equation cannot satisfy Strichartz estimates, in the whole range of Schr\"odinger admissibility.