Quantum Dynamics and generalized fractal dimensions: an introduction

Germinet, François

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2002-2003), Exposé no. 18, 14 p.

We review some recent results on quantum motion analysis, and in particular lower bounds for moments in quantum dynamics. The goal of the present exposition is to stress the role played by quantities we shall call Transport Integrals and by the so called generalized dimensions of the spectral measure in the analysis of quantum motion. We start with very simple derivations that illustrate how these quantities naturally enter the game. Then, gradually, we present successive improvements, up to most recent result.

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

[BGT1] J.-M. Barbaroux, F. Germinet, S. Tcheremchantsev, Fractal dimensions and the phenomenon of intermittency in quantum dynamics, Duke Math. J. 110, 161-193 (2001). | MR | Zbl

[BGT2] J.-M. Barbaroux, F. Germinet, S. Tcheremchantsev, Quantum diffusion and generalized fractal dimensions: the ${\mathrm{L}}^2\left({ℝ}^d\right)$ case, Actes des journées EDP de Nantes (2000). | MR

[BGT3] J.-M. Barbaroux, F. Germinet, S. Tcheremchantsev, Generalized fractal dimensions: equivalence and basic properties, J. Math. Pure et Appl. 80, 977-1012 (2001) | MR | Zbl

[BT] J.-M. Barbaroux, S. Tcheremchantsev, Universal lower bounds for quantum diffusion, J. Funct. Anal. 168, 327-354 (1999). | MR | Zbl

[BGSB] J. Bellissard, I. Guarneri, H. Schulz-Baldes, Phase-averaged transport for quasi-periodic Hamiltonians, Commun. Math. Phys. 227, 515-539 (2002). | MR | Zbl

[C] 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. | Zbl

[CM] J.M. Combes, G. Mantica, Fractal Dimensions and Quantum Evolution Associated with Sparse Potential Jacobi Matrices, Long time behaviour of classical and quantum systems (Bologna, 1999), 107-123, Ser. Concr. Appl. Math., 1, World Sci. Publishing, River Edge, NJ, 2001. | MR | Zbl

[CFKS] H. Cycon, R. Froese, W. Kirsch and B. Simon, Schrödinger Operators, Springer-Verlag, 1987. | MR | Zbl

[DT] D. Damanik and S. Tcheremchantsev, Power-law bounds on transfer matrices and quantum dynamics in one dimension, Commun. Math. Phys. 236, 513-534 (2003). | MR | Zbl

[DST] D. Damanik, A. Suto, S. Tcheremchantsev, Power-law Bounds on Transfer Matrices and Quantum Dynamics in One Dimension II, Preprint. | MR

[DBF] S. De Bièvre, G. Forni, Transport properties of kicked and quasiperiodic Hamiltonians, J. Statist. Phys. 90, 1201-1223 (1998). | MR | Zbl

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

[GKT] F. Germinet, A. Kiselev, S. Tcheremchantsev, Transfer matrices and transport for 1D Schrödinger operators with singular spectrum, Preprint.

[GK] F. Germinet, A. Klein, A characterization of the Anderson metal-insulator transport transition, Preprint. | MR

[GT] F. Germinet, S. Tcheremchantsev, Generalized fractal dimensions on the negative axis for compactly supported measures, Preprint. | MR

[G] 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, 729-733 (1993).

[GSB1] I. Guarneri and 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

[GSB2] I. Guarneri and H. Schulz-Baldes, Intermittent lower bound on quantum diffusion, Lett. Math. Phys. 49 (1999), 317-324. | MR | Zbl

[He] M. Hermann, Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d’un théorème d’Arnod et de Moser sur le tore en dimension 2, Comment. Math. Helv. 58, 453-502 (1983). | Zbl

[JSBS] S. Jitomirskaya, H. Schulz-Baldes, G. Stolz, Delocalization in random polymer models Commun. Math. Phys. 233, 27-48 (2003). | MR | Zbl

[KL] A. Kiselev and Y. Last, Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains, Duke Math. J. 102, 125-150 (2000). | MR | Zbl

[KKS] A. Klein, A. Koines, M. Seifert, Generalized Eigenfunctions for Waves in Inhomogeneous Media, J. Funct. Anal 190, 255-291 (2002). | MR | Zbl

[La] Y. Last, Quantum dynamics and decomposition of singular continuous spectrum, J. Funct. Anal. 142, 406-445 (1996). | MR | Zbl

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

[Ol] L. Olsen, A multifractal formalism, Adv. Math. 116, 82-195 (1995). | MR | Zbl

[Pe] Y. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Univ. Chicago Press, (1996). | MR | Zbl

[RS] M. Reed, B. Simon, Methods of Modern Mathematical Physics III, Scattering theory, Academic London 1979. | MR | Zbl

[Re] A. Rényi, Probability Theory, Amsterdam (1970); Calcul des Probabilités, avec un appendice sur la théorie de l’information, Dunod 1966. | Zbl

[SBB] H. Schulz-Baldes, J. Bellissard, Anomalous transport: a mathematical framework, Rev. Math. Phys. 10, 1-46 (1998). | MR | Zbl

[Si1] B. Simon, Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators, Proc. AMS 124 (1996), 3361-3369. | MR | Zbl

[St] R. Strichartz, Fourier asymptotics of fractal measures. J. Funct. Anal. 89, 154–187 (1990). | MR | Zbl

[TV] F. Takens, E. Verbitski, Multifractal analysis of local entropies for expansive homeomorphisms with specification, Commun. Math. Phys. 203, 593-612 (1999). | MR | Zbl

[Tc1] S. Tcheremchantsev, Mixed lower bounds in quantum transport, J. Funct. Anal. 197 (2003), 247-282. | MR | Zbl

[Tc2] S. Tcheremchantsev, Dynamical analysis of Schrödinger operators with sparse potentials, in preparation. | Zbl