Spectral localization estimates for abstract linear Schrödinger equations

Abstract

We study the propagation properties of abstract linear Schrödinger equations of the form $i\partial_tψ= H_0ψ+V(t)ψ$, 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 $ψ_0$ has spectral support in $(-\infty,0]$ with respect to a reference self-adjoint operator $ϕ$, then, for some $c>0$ independent of $ψ_0$ and all $t\ne0$, the solution $ψ_t$ remains spectrally supported in $(-\infty,c|t|]$ with respect to $ϕ$, up to an $O(|t|^{-n})$ remainder in norm. The main condition is that the multiple commutators of $H_0$ and $ϕ$ are uniformly bounded in operator norm up to the $(n+1)$-th order. We then apply the abstract theory to a class of nonlocal Schrödinger 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 $t\ne0$.