Quantum diffusion and generalized Rényi dimensions of spectral measures

Barbaroux, Jean-Marie; Germinet, François; Tcheremchantsev, Serguei

Journées équations aux dérivées partielles (2000), article no. 1, 16 p.

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 $\psi$ defined by $[\frac{1}{T}{\int }_0^T\parallel |X{|}^{p/2}{\mathrm{e}}^{-itH}\psi {\parallel }^2\mathrm{d}t]$. We show that this lower bound can be expressed in term of the generalized Rényi dimension of the spectral measure ${\mu }_{\psi }$ associated to the hamiltonian $H$ and the state $\psi$. We especially concentrate on continuous models.

[1] I.N. Akhiezer, I.M. Glazman, Theory of linear Operators in Hilbert spaces Vol II, Ungar, New-York, 1963.

[2] J.-M. Barbaroux, J.-M. Combes, R. Montcho, Remarks on the relation between quantum dynamics and fractal spectra, J. Math. Anal and Appl. 213 (1997), 698-722. | MR | Zbl

[3] J.-M. Barbaroux, F. Germinet, S. Tcheremchantsev, Nonlinear variation of diffusion exponents in quantum dynalics, C.R. Acad. Sci., Parist.330 Série I (1999), 409-414. Fractal dimensions and the phenomenon of intermittency in quantum dynamics, Preprint (2000). | Zbl

[4] J.-M. Barbaroux, F. Germinet, S. Tcheremchantsev, Generalized fractal dimensions : equivalences and basic properties, Preprint (2000).

[5] J.-M. Barbaroux, H. Schulz-Baldes, Anomalous quantum transport in presence of self-similar spectra, Ann. Inst. Henri Poincaré Vol 71, Numero 5 (1999), 1-21. | Numdam | MR | Zbl

[6] J.-M. Barbaroux, S. Tcheremchantsev, Universal Lower Bounds for Quantum Diffusion, J. Funct. Anal. 168 (1999), 327-354. | MR | Zbl

[7] J.-M. Combes, Connection between quantum dynamics and spectral properties of time evolution operators in «Differential Equations and Applications in Mathematical Physics», Eds. W.F. Ames, E.M. Harrel, J.V. Herod (Academic Press 1993), 59-69. | MR | Zbl

[8] J.-M. Combes, G. Mantica, A sparse potential test of Guarneri bounds, to appear in the Proceeding of the Conference on Asymptotics Properties of Time Evolutions in Classical and Quantum Systems, Bologna, 1999.

[9] Del Rio R., Jitomirskaya S., Last Y., Simon B., Operators with singular continuous spectrum IV : Hausdorff dimensions, rank one perturbations and localization, J. d'Analyse Math. 69 (1996), 153-200. | MR | Zbl

[10] K.-J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, No 85, Cambridge Univ. Press, UK, 1985. | MR | Zbl

[11] I. Guarneri, Spectral properties of quantum diffusion on discrete lattices, Europhys. Lett. 10 (1989), 95-100; On an estimate concerning quantum diffusion in the presence of a fractal spectrum, Europhys. Lett. 21 (1993), 729-733.

[12] I. Guarneri, H. Schulz-Baldes, Lower bounds on wave packet propagation by packing dimensions of spectral measures, Math. Phys. Elec. J. 5, paper 1 (1999). | MR | Zbl

[13] I. Guarneri, H. Schulz-Baldes, Intermittent lower bound on quantum diffusion, Lett. Math. Phys. 49 (1999), 317-324. | MR | Zbl

[14] S. Jitomirskaya, Y. Last : Dimensional Hausdorff properties of singular continuous spectra Phys. Rev. Letters 76 (1996), 1765-1769. | MR | Zbl

[15] Y. Last, Quantum Dynamics and Decomposition of Singular Continuous Spectrum, J. Funct. Anal 142 (1996), 406-445. | MR | Zbl

[16] G. Mantica. Quantum intermittency in almost periodic systems derived from their spectral properties, Physica D 103 (1997), 576-589; Wave Propagation in Almost-Periodic Structures, Physica D 109 (1997), 113-127. | Zbl

[17] R.S. Strichartz: Fourier asymptotics of fractal measures, J. Funct. Anal. 89 (1990), 154-187. | MR | Zbl

[18] B. Simon, Schrödinger semi-groups, Bull. Amer. Math. Soc. Vol. 7, n° 3 (1982), 447-526. | MR | Zbl

[19] S.J. Taylor, C. Tricot, Packing Measure, and its Evaluation for a Brownian Path, Trans. Amer. Math. Soc. 288 (1985), 679-699. | MR | Zbl