Homogenization of periodic Schrödinger-type equations, with lower order terms

M. A. Dorodnyi

Algebra i analiz, Tome 31 (2019) no. 6, pp. 122-196.

Résumé

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.

Détail

Keywords: periodic differential operators, Schrödinger-type equation, homogenization, effective operator, operator error estimates.

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