On complex-time heat kernels of fractional Schr\"odinger operators via Phragm\'en-Lindel\"of principle

TL;DR

This paper investigates complex-time heat kernels of fractional Schr"odinger operators with singular potentials, using a Phragm"en-Lindel"of theorem to derive spatially averaged estimates and off-diagonal bounds.

Contribution

It introduces a novel application of the Phragm"en-Lindel"of principle to analyze complex-time heat kernels of fractional Schr"odinger operators with singular potentials.

Findings

Derived spatially averaged heat kernel estimates.

Established off-diagonal bounds via interpolation.

Applied Phragm"en-Lindel"of theorem to complex analysis of operators.

Abstract

We consider fractional Schr\"odinger operators with possibly singular potentials and derive certain spatially averaged estimates for its complex-time heat kernel. The main tool is a Phragm\'en-Lindel\"of theorem for polynomially bounded functions on the right complex half-plane. The interpolation leads to possibly nonoptimal off-diagonal bounds.