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{\Large\bf Multiresolution in Quantum Dynamics: From Localization to
Pattern Formation}\\
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{\large\bf Antonina Fedorova, Michael Zeitlin}\\
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Institute of Problems of Mechanical Engineering \\
(IPME RAS) Russian Academy of Sciences\\
Russia, 199178, St. Petersburg, V.O., Bolshoj pr., 61\\
E-mail : anton@math.ipme.ru, zeitlin@math.ipme.ru\\
http://www.ipme.nw.ru/zeitlin.html\\
http://www.ipme.ru/zeitlin.html\\

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{\large\bf Abstract}
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We present the application of variational-wavelet analysis
to numerical-analytical calculations of Wigner functions
in (nonlinear) quasiclassical dynamical problems as solution of
corresponding (pseudo)differential Wigner-von Neumann equations [1],[2].
(Naive) deformation quantization and multiresolution
representations are the key points.
We construct the representations via multiscale expansions in generalized
coherent states or high-localized nonlinear eigenmodes in the base of
compactly supported wavelets and wavelet packets which
are natural nonlinear generalization of standard coherent, squeezed,
thermal squeezed states
corresponding only to quadratical systems (linear dynamics) with
Gaussian Wigner functions.
As result we calculate quantum corrections to classical dynamics
described by arbitrary nonlinear polynomial
Hamiltonians.
We give contributions to our full quasiclassical representation
from each level of underlying hidden internal scales [3].
We consider applications of constructed localized and pattern-like
solutions to dynamics of decoherence, entangled states etc.
Related applications are considered in the second abstract submitted.


\begin{thebibliography}{3}
\bibitem{1}
 A. Fedorova, M. Zeitlin, Quasiclassical Calculations for Wigner
Functions via Multiresolution,
arXiv preprint: physics/0101006 and World Scientific, 2002, in
press
\bibitem{2}
 A. Fedorova, M. Zeitlin, Quasiclassical Calculations of Wigner
Functions in Nonlinear Beam
Dynamics,
Proceedings PAC2001, pp. 1814-1816, IEEE, 2002;
http://accelconf.web.cern.ch/AccelConf/p01/PAPERS/TPPH066.pdf
or arXiv preprint: physics/0106005
 \bibitem{3}
A. Fedorova, M. Zeitlin, Fast Calculation Methods
in Collective Dynamical Models of Beam/Plasma Physics,
arXiv preprint: physics/0206050

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