Homogenization of nonstationary Schrödinger type equations with periodic coefficients

2015/08/30 by T. A. Suslina, Tatiana Suslina, Suslina, Tatiana
Computer Science · Engineering · Mathematics · #35B27 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations #math.AP #msc:35B27

paper · pdf · doi:10.48550/arxiv.1508.07641

78 pages

arxiv created 2015/08/30 · openalex publication_date 2015/08/30 · arxiv updated 2015/09/01 · openalex created_date 2022/09/05 · openalex updated_date 2026/07/28

Abstract

In L₂(ℝ^d;\mathbb Cⁿ) we consider selfadjoint strongly elliptic second order differential operators \mathcal A_ε with periodic coefficients depending on \mathbf x/ε. We study the behavior of the operator exponential exp(-i \mathcal A_ε τ), τ∈ \mathbb R, for small ε. Approximations for this exponential in the (H^s→ L₂)-operator norm with a suitable s are obtained. The results are applied to study the behavior of the solution \mathbf u_ε of the Cauchy problem for the Schrödinger type equation i ∂_τ\mathbf u_ε = \mathcal A_ε \mathbf u_ε.

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