Error Estimates of the Bloch Band-Based Gaussian Beam Superposition for the Schr\"odinger Equation

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摘要：

This work is concerned with asymptotic approximations of the semi-classicalSchr\"odinger equation in periodic media using Gaussian beams. For theunderlying equation, subject to a highly oscillatory initial data, a hybrid ofthe Gaussian beam approximation and homogenization leads to the Blocheigenvalue problem and associated evolution equations for Gaussian beamcomponents in each Bloch band. We formulate a superposition of Bloch-band basedGaussian beams to generate high frequency approximate solutions to the originalwave field. For initial data of a sum of finite number of band eigen-functions,we prove that the first-order Gaussian beam superposition converges to theoriginal wave field at a rate of $\epsilon^{1/2}$, with $\epsilon$ thesemiclassically scaled constant, as long as the initial data for Gaussian beamcomponents in each band are prepared with same order of error or smaller. For anatural choice of initial approximation, a rate of $\epsilon^{1/2}$ of initialerror is verified.

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关键词：

Mathematics - Analysis of PDEs

DOI：

10.1090/conm/640/12852

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年份：

2014