Spectral localization estimates for abstract linear Schr\"odinger equations

TL;DR

This paper establishes spectral localization propagation estimates for abstract linear Schrödinger equations, providing explicit conditions under which initial spectral support remains localized within a linearly expanding region over time.

Contribution

It introduces explicit sufficient conditions based on boundedness of multiple commutators for spectral support propagation in abstract Schrödinger equations.

Findings

Spectral support propagates linearly with time under certain commutator bounds.

Solutions with compact initial support remain approximately supported in a linearly expanding region.

Provides polynomial decay estimates for the spectral tail outside the main support.

Abstract

We study the propagation properties of abstract linear Schr\"odinger equations of the form $i\partial_t\psi = H_0\psi+V(t)\psi$, where $H_0$ is a self-adjoint operator and $V(t)$ a time-dependent potential. We present explicit sufficient conditions ensuring that if the initial state $\psi_0$ has spectral support in $(-\infty,0]$ with respect to a reference self-adjoint operator $\phi$, then, for some $c>0$ independent of $\psi_0$ and all $t\ne0$, the solution $\psi_t$ remains spectrally supported in $(-\infty,c|t|]$ with respect to $\phi$, up to an $O(|t|^{-n})$ remainder in norm. The main condition is that the multiple commutators of $H_0$ and $\phi$ are uniformly bounded in operator norm up to the $(n+1)$-th order. We then apply the abstract theory to a class of nonlocal Schr\"odinger equations on $\mathbb{R}^d$, proving that any solution with compactly supported initial state remains approximately supported, up to a polynomially suppressed tail in $L^2$-norm, inside a linearly spreading region around the initial support for all $t\ne0$.