Sharp $L^p$ estimates for Schr\"odinger groups

TL;DR

This paper establishes sharp $L^p$ bounds for Schr"odinger groups under certain decay conditions on the heat operator, extending results to operators with magnetic and electric potentials.

Contribution

It proves new sharp $L^p$ estimates for Schr"odinger groups based on off-diagonal decay of the heat operator, applicable to magnetic and electric potential operators.

Findings

Sharp $L^p$ bounds for Schr"odinger groups established.

Results apply to operators with magnetic and electric potentials.

Extends previous bounds to broader class of operators.

Abstract

Consider a non-negative self-adjoint operator $H$ in $L^2(\mathbb{R}^d)$. We suppose that its heat operator $e^{-tH}$ satisfies an off-diagonal algebraic decay estimate, for some exponents $p_0\in[0,2)$. Then we prove sharp $L^p\to L^p$ frequency truncated estimates for the Schr\"odinger group $e^{itH}$ for $p\in[p_0,p'_0]$. In particular, our results apply to every operator of the form $H=(i\nabla+A)^2+V$, with a magnetic potential $A\in L^2_{loc}(\mathbb{R}^d,\mathbb{R}^d)$ and an electric potential $V$ whose positive and negative parts are in the local Kato class and in the Kato class, respectively.