Operator error estimates for homogenization of the nonstationary\n Schr "odinger-type equations: sharpness of the results

2020/05/12 by Mark Dorodnyi, Dorodnyi, Mark · 1 citation
Computer Science · Engineering · Mathematics · #35B27 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2005.06516

openalex publication_date 2020/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In L₂ (\ℝ^d; \ℂⁿ), we consider a selfadjoint matrix\nstrongly elliptic second order differential operator \A_\ε\nwith periodic coefficients depending on \x/\ε. We find\napproximations of the exponential e^{-i \τ \A_\ε}, \τ\n\∈ \ℝ, for small \ε in the (H^s \→ L₂)-operator norm\nwith suitable s. The sharpness of the error estimates with respect to \τ\nis discussed. The results are applied to study the behavior of the solution\n\u_\ε of the Cauchy problem for the Schr "odinger-type\nequation i\∂_\τ \u_\ε = \A_\ε\n\u_\ε + \F.\n

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