文摘
In k to view the MathML source">L2(Rd;Cn), we consider selfadjoint strongly elliptic second order differential operators k to view the MathML source">Aε with periodic coefficients depending on k to view the MathML source">x/ε. We study the behavior of the operator k to view the MathML source">exp(−iAετ), k to view the MathML source">τ∈R, for small ε . Approximations for this exponential in the k to view the MathML source">(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 k to view the MathML source">uε of the Cauchy problem for the Schrödinger-type equation k to view the MathML source">i∂τuε=Aεuε+F. Applications to the nonstationary Schrödinger equation and the two-dimensional Pauli equation with singular rapidly oscillating potentials are given.