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Semiclassical Wigner function and geometrical optics. (English) Zbl 1038.81039

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.

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
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