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On the evolution of the semi-classical Wigner function in higher dimensions

Published online by Cambridge University Press:  21 October 2005

S. FILIPPAS  and
G. N. MAKRAKIS
S. FILIPPAS
Affiliation:
Department of Applied Mathematics, University of Crete, 71409 Heraklion, Greece Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion, Greece Email: [email protected], [email protected]
G. N. MAKRAKIS
Affiliation:
Department of Applied Mathematics, University of Crete, 71409 Heraklion, Greece Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion, Greece Email: [email protected], [email protected]
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Abstract

The limit Wigner measure of a WKB function satisfies a simple transport equation in phase-space and is well suited for capturing oscillations at scale of order $O(\epsilon)$, but it fails, for instance, to provide the correct amplitude on caustics where different scales appear. We define the semi-classical Wigner function of an $N$-dimensional WKB function, as a suitable formal approximation of its scaled Wigner function. The semi-classical Wigner function is an oscillatory integral that provides an $\epsilon$-dependent regularization of the limit Wigner measure, it obeys a transport-dispersive evolution law in phase space, and it is well defined even at simple caustics.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 17 , Issue 1 , February 2006 , pp. 33 - 62
Copyright
2005 Cambridge University Press

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