Spectral localization estimates for abstract linear Schr\"odinger equations

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 $t\ne0$.