Global geometrical optics method for vector-valued Schrödinger problems

doi:10.1007/s11464-018-0704-1

Front. Math. China

2018, Vol. 13

Issue (3) : 579-606 https://doi.org/10.1007/s11464-018-0704-1

RESEARCH ARTICLE

Global geometrical optics method for vector-valued Schrödinger problems

Jiashun HU(

), Xiang MA, Chunxiong ZHENG

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

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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.

Keywords Global geometrical optics method Hamiltonian system unitary representation caustics semiclassical approximation

Corresponding Author(s): Jiashun HU

Issue Date: 11 June 2018

Cite this article:

Jiashun HU,Xiang MA,Chunxiong ZHENG. Global geometrical optics method for vector-valued Schrödinger problems[J]. Front. Math. China, 2018, 13(3): 579-606.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-018-0704-1
https://academic.hep.com.cn/fmc/EN/Y2018/V13/I3/579

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