Semiclassical Wigner Function and Geometrical Optics

Abstract

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 geometrical 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.