On the fractional powers of a Schr\"odinger operator with a Hardy-type potential

TL;DR

This paper investigates the unique continuation and asymptotic behavior of fractional Schr"odinger operators with Hardy potentials using advanced mathematical tools like Almgren monotonicity and blow-up analysis.

Contribution

It introduces new techniques for analyzing fractional Schr"odinger operators with Hardy potentials, providing insights into their unique continuation properties and asymptotic profiles.

Findings

Established strong unique continuation properties.

Classified asymptotic profiles of solutions.

Developed an Almgren monotonicity formula for this context.

Abstract

Strong unique continuation properties and a classification of the asymptotic profiles are established for the fractional powers of a Schr\"odinger operator with a Hardy-type potential, by means of an Almgren monotonicity formula combined with a blow-up analysis.