Filippas, Stathis; Makrakis, George N. Semiclassical Wigner function and geometrical optics. (English) Zbl 1038.81039 Multiscale Model. Simul. 1, No. 4, 674-710 (2003). Summary: We consider the problem of high-frequency asymptotics for the time-dependent one-dimensional Schrödinger equation with rapidly oscillating initial data. This problem is commonly studied via the WKB method. An alternative method is based on the limit Wigner measure. This approach recovers geometric optics, but, like the WKB method, it fails at caustics. To remedy this deficiency we employ the semiclassical Wigner function which is a formal asymptotic approximation of the scaled Wigner function but also a regularization of the limit Wigner measure. We obtain Airy-type asymptotics for the semiclassical Wigner function. This representation is shown to be exact in the context of concrete examples. In these examples we compute both the semiclassical and the limit Wigner function, as well as the amplitude of the wave field near a fold or a cusp caustic, which evolve naturally from suitable initial data. Cited in 5 Documents MSC: 81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics 81V80 Quantum optics 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory 78A05 Geometric optics 58K55 Asymptotic behavior of solutions to equations on manifolds 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations 34E05 Asymptotic expansions of solutions to ordinary differential equations 34E13 Multiple scale methods for ordinary differential equations Keywords:high-frequency asymptotics; oscillating initial data; limit Wigner measure; caustics; Airy-type asymptotics PDFBibTeX XMLCite \textit{S. Filippas} and \textit{G. N. Makrakis}, Multiscale Model. Simul. 1, No. 4, 674--710 (2003; Zbl 1038.81039) Full Text: DOI