Homogenization of periodic Schrödinger-type equations, with lower order terms
Mark Aleksandrovich Dorodnyi · St Petersburg Mathematical Journal · 2020
In $L_2 (\mathbb {R}^d; \mathbb {C}^n)$, consider a selfadjoint matrix elliptic second order differential operator $\mathcal {B}_\varepsilon$, $0<\varepsilon \leq 1$, with periodic coefficients depending on $\mathbf {x}/\varepsilon$. The principal part of the operator is given in a factorized form, the operator involves first and zero order terms. Approximation is found for the operator exponential $e^{-is \mathcal {B}_\varepsilon }$, $s \in \mathbb {R}$, for small $\varepsilon$ in the ($H^r \to L_2$)-operator norm with a suitable $r$. The results are applied to study the behavior of the solution $\mathbf {u}_\varepsilon$ of the Cauchy problem for the nonstationary Schrödinger-type equation $i\partial _{s} \mathbf {u}_\varepsilon = \mathcal {B}_\varepsilon \mathbf {u}_\varepsilon + \mathbf {F}$. Applications to the magnetic Schrödinger equation and the two-dimensional Pauli equation with singular potentials are considered.