Commun. Comput. Phys., 11 (2012), pp. 951-984. |
Composite Coherent States Approximation for One-Dimensional Multi-Phased Wave Functions Dongsheng Yin ^{1}, Chunxiong Zheng ^{1*} 1 Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P.R. China.Received 10 October 2010; Accepted (in revised version) 25 May 2011 Available online 18 November 2011 doi:10.4208/cicp.101010.250511a Abstract The coherent states approximation for one-dimensional multi-phased wave functions is considered in this paper. The wave functions are assumed to oscillate on a characteristic wave length $O(\epsilon)$ with $\epsilon\ll 1$. A parameter recovery algorithm is first developed for single-phased wave function based on a moment asymptotic analysis. This algorithm is then extended to multi-phased wave functions. If cross points or caustics exist, the coherent states approximation algorithm based on the parameter recovery will fail in some local regions. In this case, we resort to the windowed Fourier transform technique, and propose a composite coherent states approximation method. Numerical experiments show that the number of coherent states derived by the proposed method is much less than that by the direct windowed Fourier transform technique. AMS subject classifications: 33F05, 68W25Key words: Coherent state, high frequency, windowed Fourier transform. *Corresponding author. Email: dyin@math.tsinghua.edu.cn (D. Yin), czheng@math.tsinghua.edu.cn (C. Zheng) |