Regularity of hyperbolic magnetic Schr\"odinger equation with oscillating coefficients

TL;DR

This paper investigates the regularity properties of the hyperbolic magnetic Schrödinger equation with oscillating and singular coefficients, establishing bounds on regularity loss using microlocal analysis and counterexamples.

Contribution

It provides new insights into the regularity behavior of the equation with oscillating coefficients, including optimal bounds and counterexamples demonstrating their sharpness.

Findings

Upper bound of regularity loss established

Counterexample confirms the optimality of the bounds

Techniques from microlocal and harmonic analysis are employed

Abstract

This paper mainly discuss the regularity behavior of the hyperbolic magnetic Schroedinger equation with singular coefficients near the origin. We apply the techniques from the microlocal analysis to explore the upper bound of loss of regularity. Furthermore, in order to demonstrate the optimality of the result, a delicate counterexample with periodic coefficients will be constructed to show the lower bound of loss of regularity by the application of harmonic analysis and instability arguments.