On the fractional Sch\"odinger equation with variable coefficients

TL;DR

This paper investigates the initial value problem for a semi-linear fractional Schrödinger equation with variable coefficients, establishing well-posedness and unique continuation properties through analysis of the associated anisotropic fractional elliptic operator.

Contribution

It introduces new results on local well-posedness and unique continuation for fractional Schrödinger equations with variable coefficients, expanding understanding of their mathematical properties.

Findings

Established local well-posedness of the IVP.

Proved unique continuation results for solutions.

Analyzed properties of the anisotropic fractional elliptic operator.

Abstract

We study the initial value problem (IVP) associated to the semi-linear fractional Sch\"odinger equation with variable coefficients. We deduce several properties of the anisotropic fractional elliptic operator modelling the dispersion relation and use them to establish the local well-posedness for the corresponding IVP. Also, we obtain unique continuation results concerning the solutions of this problem. These are consequences of uniqueness properties that we prove for the fractional elliptic operator with variable coefficients