Spectral approach to homogenization of nonstationary Schrödinger-type equations

In L2(Rd;Cn)L2(Rd;Cn), we consider selfadjoint strongly elliptic second order differential operators AεAε with periodic coefficients depending on x/εx/ε. We study the behavior of the operator exp⁡(−iAετ)exp⁡(−iAετ), τ∈Rτ∈R, for small ε  . Approximations for this exponential in the (Hs→L2)(Hs→L2)-operator norm are obtained. The method is based on the scaling transformation, the Floquet–Bloch theory, and the analytic perturbation theory. The results are applied to study the behavior of the solution uεuε of the Cauchy problem for the Schrödinger-type equation i∂τuε=Aεuε+Fi∂τuε=Aεuε+F. Applications to the nonstationary Schrödinger equation and the two-dimensional Pauli equation with singular rapidly oscillating potentials are given.