Improved time-decay for a class of scaling critical electromagnetic Schr\"odinger flows

TL;DR

This paper establishes improved decay estimates for electromagnetic Schrödinger flows with critical scaling, showing how weighted norms decay faster over time, depending on the angular operator's ground level.

Contribution

It introduces a novel weighted decay estimate for electromagnetic Schrödinger flows with critical scaling, linking decay rates to the angular operator's spectral properties.

Findings

Weighted decay estimates depend on the angular operator's ground level.

Results apply to both Schrödinger and heat semigroups.

Decay rates are explicitly quantified in terms of space dimension and spectral data.

Abstract

We consider a Schr\"odinger hamiltonian $H(A,a)$ with scaling critical and time independent external electromagnetic potential, and assume that the angular operator $L$ associated to $H$ is positive definite. We prove the following: if $\|e^{-itH(A,a)}\|_{L^1\to L^\infty}\lesssim t^{-n/2}$, then $ \||x|^{-g(n)}e^{-itH(A,a)}|x|^{-g(n)}\|_{L^1\to L^\infty}\lesssim t^{-n/2-g(n)}$, $g(n)$ being a positive number, explicitly depending on the ground level of $L$ and the space dimension $n$. We prove similar results also for the heat semi-group generated by $H(A,a)$.