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# Communications in Mathematical Sciences

## Volume 10 (2012)

### Number 4

### Gaussian beam methods for the Dirac equation in the semi-classical regime

Pages: 1301 – 1315

(Fast Communication)

DOI: http://dx.doi.org/10.4310/CMS.2012.v10.n4.a14

#### Authors

#### Abstract

The Dirac equation is an important model in relativistic quantum mechanics. In the semi-classical regime ε≪1, even a spatially spectrally accurate time splitting method [6] requires the mesh size to be O(ε), which makes the direct simulation extremely expensive. In this paper, we present the Gaussian beam method for the Dirac equation. With the help of an eigenvalue decomposition, the Gaussian beams can be independently evolved along each eigenspace and summed to construct an approximate solution of the Dirac equation. Moreover, the proposed Eulerian Gaussian beam keeps the advantages of constructing the Hessian matrices by simply using the derivatives of level set functions. Finally, several numerical examples show the efficiency and accuracy of the method.

#### Keywords

Dirac equation, semi-classical regime, Gaussian beam method, Lagrangian and Eulerian formulations

#### 2010 Mathematics Subject Classification

65Mxx, 81Q05, 81Q20