Landis-type conjecture for the half-Laplacian

TL;DR

This paper proves that solutions to a fractional Schrödinger equation with drift and potential, which decay exponentially, must be trivial, advancing understanding of unique continuation properties for fractional PDEs.

Contribution

It establishes a Landis-type conjecture for the half-Laplacian, demonstrating triviality of solutions with exponential decay using novel analytical techniques.

Findings

Solutions with exponential decay are trivial

Utilizes Caffarelli-Silvestre extension and Liouville-type theorem

Advances unique continuation theory for fractional PDEs

Abstract

In this paper, we study the Landis-type conjecture, i.e., unique continuation property from infinity, of the fractional Schr\"{o}dinger equation with drift and potential terms. We show that if any solution of the equation decays at a certain exponential rate, then it must be trivial. The main ingredients of our proof are the Caffarelli-Silvestre extension and Armitage's Liouville-type theorem.