Global geometrical optics method for vector-valued Schrödinger problems

Abstract

We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11(1): 105-140], to the more general vectorvalued Schrödinger problems in the semi-classical regime. The key ingredient in the global geometrical optics method is a moving frame technique in the phase space. The governing equation is transformed into a new equation but of the same type when expressed in any moving frame induced by the underlying Hamiltonian ow. The classical Wentzel-Kramers-Brillouin (WKB) analysis benefits from this treatment as it maintains valid for arbitrary but fixed evolutionary time. It turns out that a WKB-type function defined merely on the underlying Lagrangian submanifold can be obtained with the help of this moving frame technique, and from which a uniform first-order approximation of the wave field can be derived, even around caustics. The general theory is exemplified by two specific instances. One is the two-level Schrödinger system and the other is the periodic Schrödinger equation. Numerical tests validate the theoretical results.

Key words： Global geometrical optics method; Hamiltonian system; unitary representation; caustics; semiclassical approximation

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