Spectral localization estimates for abstract linear Schrödinger equations

We study the propagation properties of abstract linear Schrödinger equations of the form $i{\mathrm{\partial }}_t\psi =H_0\psi +V(t)\psi$, 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 $(-\mathrm{\infty },0]$ with respect to a reference self-adjoint operator $\phi$, then, for some $c>0$ independent of ${\psi }_0$ and all $t\mathrm{\ne }0$, the solution ${\psi }_t$ remains spectrally supported in $(-\mathrm{\infty },c\mathrm{\mid }t\mathrm{\mid }]$ with respect to $\phi$, up to an $O(\mathrm{\mid }t{\mathrm{\mid }}^{-n})$ 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 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 $t\mathrm{\ne }0$.