Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  develop a gauge-invariant frozen Gaussian approximation (GIFGA) method for the Schrodinger 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.  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].
['Ricardo Delgadillo', 'Jianfeng Lu', 'Xu Yang']
I am writing a research paper on a idea Gauge-invariant frozen Gaussian approximation method for the Schr\"odinger equation with periodic potentials
[{"from": "human", "value": "I am writing a research paper on a idea Gauge-invariant frozen Gaussian approximation method for the Schr\\\"odinger equation with periodic potentials"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  develop a gauge-invariant frozen Gaussian approximation (GIFGA) method for the Schrodinger 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.  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]."}]