Displaying similar documents to “Quantum Dynamics and generalized fractal dimensions: an introduction”

Transfer matrices and transport for Schrödinger operators

François Germinet, Alexander Kiselev, Serguei Tcheremchantsev (2004)

Annales de l’institut Fourier

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We provide a general lower bound on the dynamics of one dimensional Schrödinger operators in terms of transfer matrices. In particular it yields a non trivial lower bound on the transport exponents as soon as the norm of transfer matrices does not grow faster than polynomially on a set of energies of full Lebesgue measure, and regardless of the nature of the spectrum. Applications to Hamiltonians with a) sparse, b) quasi-periodic, c) random decaying potential...

Quantum diffusion and generalized Rényi dimensions of spectral measures

Jean-Marie Barbaroux, François Germinet, Serguei Tcheremchantsev (2000)

Journées équations aux dérivées partielles

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We estimate the spreading of the solution of the Schrödinger equation asymptotically in time, in term of the fractal properties of the associated spectral measures. For this, we exhibit a lower bound for the moments of order p at time T for the state ψ defined by [ 1 T ∫ 0 T ∥ | X | p / 2 e - i t H ψ ∥ 2 d t ] . We show that this lower bound can be expressed in term of the generalized Rényi dimension of the spectral measure μ ψ associated to the hamiltonian H and the state ψ . We especially concentrate on continuous models. ...