Wavelet dimensions and time evolution

Abstract : In this chapter, we study some aspects of the chaotic behaviour of the time evolution generated by Hamiltonian systems, or more generally dynamical systems. We introduce a characteristic quantity, namely the lacunarity dimension, to quantify the intermittency phenomena that can arise in the time evolution. We then focus on the time evolution of wave packets according to the Schrödinger equation with time independent Hamiltonian. We introduce a set of fractal dimensions constructed by means of the wavelet transform, the (generalized) wavelet dimensions. We show that the lacunarity dimension of the wave packets can be obtained via the wavelet dimensions of the spectral measure of the Schrödinger operator. This establishes a precise link between the long time chaotic behaviour of the wave packets and the small scales spectral properties of the Hamiltonian.
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Charles-Antoine Guérin, Matthias Holschneider. Wavelet dimensions and time evolution. J. C. van den Berg Wavelets in Physics, Cambridge University Press, pp.421-448, 1999, 9780521593113. ⟨10.1017/CBO9780511613265.014⟩. ⟨hal-00083218⟩

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