Gauge-invariant frozen Gaussian approximation method for the schrödinger equation with periodic potentials

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© 2016 Society for Industrial and Applied Mathematics.We develop a gauge-invariant frozen Gaussian approximation (GIFGA) method for the Schrödinger equation (LSE) with periodic potentials in the semiclassical regime. The method generalizes the Herman-Kluk propagator for LSE to the case with periodic media. It provides an efficient computational tool based on asymptotic analysis on phase space and Bloch waves to capture the high-frequency oscillations of the solution. Compared to geometric optics and Gaussian beam methods, GIFGA works in both scenarios of caustics and beam spreading. Moreover, it is invariant with respect to the gauge choice of the Bloch eigenfunctions and thus avoids the numerical difficulty of computing gauge-dependent Berry phase. We numerically test the method by several one-dimensional examples; in particular, the first order convergence is validated, which agrees with our companion analysis paper [Frozen Gaussian Approximation for High Frequency Wave Propagation in Periodic Media, arXiv:1504.08051, 2015].






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Delgadillo, R, J Lu and X Yang (2016). Gauge-invariant frozen Gaussian approximation method for the schrödinger equation with periodic potentials. SIAM Journal on Scientific Computing, 38(4). pp. A2440–A2463. 10.1137/15M1040384 Retrieved from

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Jianfeng Lu

Professor of Mathematics

Jianfeng Lu is an applied mathematician interested in mathematical analysis and algorithm development for problems from computational physics, theoretical chemistry, materials science, machine learning, and other related fields.

More specifically, his current research focuses include:
High dimensional PDEs; generative models and sampling methods; control and reinforcement learning; electronic structure and many body problems; quantum molecular dynamics; multiscale modeling and analysis.

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