34 research outputs found

    Irreversible quantum graphs

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    Irreversibility is introduced to quantum graphs by coupling the graphs to a bath of harmonic oscillators. The interaction which is linear in the harmonic oscillator amplitudes is localized at the vertices. It is shown that for sufficiently strong coupling, the spectrum of the system admits a new continuum mode which exists even if the graph is compact, and a {\it single} harmonic oscillator is coupled to it. This mechanism is shown to imply that the quantum dynamics is irreversible. Moreover, it demonstrates the surprising result that irreversibility can be introduced by a "bath" which consists of a {\it single} harmonic oscillator

    Spectral and Localization Properties for the One-Dimensional Bernoulli Discrete Dirac Operator

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    A 1D Dirac tight-binding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schr?odinger model. For random Bernoulli potentials taking two values (without correlations), for typical realizations and for all values of the mass, it is shown that its spectrum is pure point, whereas the zero mass case presents dynamical delocalization for specific values of the energy. The massive case presents dynamical localization (excluding some particular values of the energy). Finally, for general potentials the dynamical moments for distinct masses are compared, especially the massless and massive Bernoulli cases.Comment: no figure; 24 pages; to appear in Journal of Mathematical Physic

    Linear Statistics of Point Processes via Orthogonal Polynomials

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    For arbitrary ÎČ>0\beta > 0, we use the orthogonal polynomials techniques developed by R. Killip and I. Nenciu to study certain linear statistics associated with the circular and Jacobi ÎČ\beta ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these results are new.Comment: Added references, corrected typos. To appear, J. Stat. Phy

    CMV matrices in random matrix theory and integrable systems: a survey

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    We present a survey of recent results concerning a remarkable class of unitary matrices, the CMV matrices. We are particularly interested in the role they play in the theory of random matrices and integrable systems. Throughout the paper we also emphasize the analogies and connections to Jacobi matrices.Comment: Based on a talk given at the Short Program on Random Matrices, Random Processes and Integrable Systems, CRM, Universite de Montreal, 200

    Canonical moments and random spectral measures

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    We study some connections between the random moment problem and the random matrix theory. A uniform draw in a space of moments can be lifted into the spectral probability measure of the pair (A,e) where A is a random matrix from a classical ensemble and e is a fixed unit vector. This random measure is a weighted sampling among the eigenvalues of A. We also study the large deviations properties of this random measure when the dimension of the matrix grows. The rate function for these large deviations involves the reversed Kullback information.Comment: 32 pages. Revised version accepted for publication in Journal of Theoretical Probabilit

    Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I

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    Classical Schur analysis is intimately connected to the theory of orthogonal polynomials on the circle [Simon, 2005]. We investigate here the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szeg\H o theory, in the case where the interpolation points may accumulate on the unit circle. This leads us to generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields asymptotics of a novel type.Comment: a preliminary version, 39 pages; some changes in the Introduction, Section 5 (Szeg\H o type asymptotics) is extende

    Generalized eigenvalue-counting estimates for the Anderson model

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    We generalize Minami's estimate for the Anderson model and its extensions to nn eigenvalues, allowing for nn arbitrary intervals and arbitrary single-site probability measures with no atoms. As an application, we derive new results about the multiplicity of eigenvalues and Mott's formula for the ac-conductivity when the single site probability distribution is H\"older continuous.Comment: Minor revisio

    Variational Estimates for Discrete Schrödinger Operators with Potentials of Indefinite Sign

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    Let H be a one-dimensional discrete Schrödinger operator. We prove that if σess(H) ⊂ [−2, 2], then H − H0 is compact and σess(H) = [−2, 2]. We also prove that if H0 + 1 4 VÂČ has at least one bound state, then the same is true for H0 + V. Further, if H0 + 1 4 VÂČ has infinitely many bound states, then so does H0 +V. Consequences include the fact that for decaying potential V with lim inf|n|→ ∞ |nV (n) |> 1, H0 + V has infinitely many bound states; the signs of V are irrelevant. Higher-dimensional analogues are also discussed
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