Displaying similar documents to “Quantum diffusion and generalized Rényi dimensions of spectral measures”

Quantum Dynamics and generalized fractal dimensions: an introduction

François Germinet (2002-2003)

Séminaire Équations aux dérivées partielles

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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 and by the so called 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.

Accurate Spectral Asymptotics for periodic operators

Victor Ivrii (1999)

Journées équations aux dérivées partielles

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Asymptotics with sharp remainder estimates are recovered for number 𝐍 ( τ ) of eigenvalues of operator A ( x , D ) - t W ( x , x ) crossing level E as t runs from 0 to τ , τ → ∞ . Here A is periodic matrix operator, matrix W is positive, periodic with respect to first copy of x and decaying as second copy of x goes to infinity, E either belongs to a spectral gap of A or is one its ends. These problems are first treated in papers of M. Sh. Birman, M. Sh. Birman-A. Laptev and M. Sh. Birman-T. Suslina.

Formulae for joint spectral radii of sets of operators

Victor S. Shulman, Yuriĭ V. Turovskii (2002)

Studia Mathematica

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The formula ϱ ( M ) = m a x ϱ χ ( M ) , r ( M ) is proved for precompact sets M of weakly compact operators on a Banach space. Here ϱ(M) is the joint spectral radius (the Rota-Strang radius), ϱ χ ( M ) is the Hausdorff spectral radius (connected with the Hausdorff measure of noncompactness) and r(M) is the Berger-Wang radius.

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...

Quotient of spectral radius, (signless) Laplacian spectral radius and clique number of graphs

Kinkar Ch. Das, Muhuo Liu (2016)

Czechoslovak Mathematical Journal

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In this paper, the upper and lower bounds for the quotient of spectral radius (Laplacian spectral radius, signless Laplacian spectral radius) and the clique number together with the corresponding extremal graphs in the class of connected graphs with n vertices and clique number ω ( 2 ≤ ω ≤ n ) are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved.