RESEARCH ARTICLE

Global geometrical optics method for vector-valued Schrödinger problems

  • Jiashun HU ,
  • Xiang MA ,
  • Chunxiong ZHENG
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  • Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received date: 15 May 2017

Accepted date: 10 Jul 2017

Published date: 11 Jun 2018

Copyright

2018 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature

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.

Cite this article

Jiashun HU , Xiang MA , Chunxiong ZHENG . Global geometrical optics method for vector-valued Schrödinger problems[J]. Frontiers of Mathematics in China, 2018 , 13(3) : 579 -606 . DOI: 10.1007/s11464-018-0704-1

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