# Progressive intrinsic ultracontractivity and heat kernel estimates for non-local Schr\"odinger operators

## TL;DR

This paper investigates the long-time behavior of semigroups generated by non-local Schr"odinger operators, establishing sharp heat kernel estimates and introducing the concept of progressive intrinsic ultracontractivity.

## Contribution

It introduces the new regularity property of progressive intrinsic ultracontractivity and provides sharp two-sided heat kernel estimates for a broad class of non-local Schr"odinger operators.

## Key findings

- Established sharp two-sided heat kernel estimates for large times
- Identified and characterized progressive intrinsic ultracontractivity
- Applied results to heat trace and heat content analysis

## Abstract

We study the long-time asymptotic behaviour of semigroups generated by non-local Schr\"odinger operators of the form $H = -L+V$; the free operator $L$ is the generator of a symmetric L\'evy process in $\mathbb R^d$, $d > 1$ (with non-degenerate jump measure) and $V$ is a sufficiently regular confining potential. We establish sharp two-sided estimates of the corresponding heat kernels for large times and identify a new general regularity property, which we call progressive intrinsic ultracontractivity, to describe the large-time evolution of the corresponding Schr\"odinger semigroup. We discuss various examples and applications of these estimates, for instance we characterize the heat trace and heat content. Our examples cover a wide range of processes and we have to assume only mild restrictions on the growth, resp.\ decay, of the potential and the jump intensity of the free process. Our approach is based on a combination of probabilistic and analytic methods; our examples include fractional and quasi-relativistic Schr\"odinger operators.