Jonathan Breuer

Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees

We present examples of rooted tree graphs for which the Laplacian has singular continuous spectral measures. For some of these examples we further establish fractional Hausdorff dimensions. The singular continuous components, in these models, have an interesting multiplicity structure. The results are obtained via a decomposition of the Laplacian into a direct sum of Jacobi matrices.

Singular spectrum for radial trees

We prove several results showing that absolutely continuous spectrum for the Laplacian on radial trees is a rare event. In particular, we show that metric trees with unbounded edges have purely singular spectrum and that, generically (in the sense of Baire), radial trees have purely singular continuous spectrum.

Universality of mesoscopic fluctuations for orthogonal polynomial ensembles

AbstractWe prove that the fluctuations of mesoscopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green’s function for the associated Jacobi matrices.As a particular consequencewe obtain a central limit theorem for the modified Jacobi Unitary Ensembles on all mesoscopic scales.

Sine kernel asymptotics for a class of singular measures

Abstract We construct a family of measures on R that are purely singular with respect to the Lebesgue measure, and yet exhibit universal sine kernel asymptotics in the bulk. The measures are best described via their Jacobi recursion coefficients: these are sparse perturbations of the recursion coefficients corresponding to Chebyshev polynomials of the second kind. We prove convergence of the renormalized Christoffel–Darboux kernel to the sine kernel for any sufficiently sparse decaying perturbation.

Continuity in symmetry and in distinguishability of states: The symmetry numbers of nonrigid molecules

We establish a connection between the concept of distinguishability of quantum states and the concept of continuity in geometric symmetry. For this purpose, we employ the continuous symmetry measure and the nuclear wave functions of a molecule, and evaluate the physical effects of deviation from geometric symmetry. We apply this tool in presenting a unified approach to assigning symmetry numbers to rigid and nonrigid molecules, and readdress, as a specific example, the evaluation of the temperature dependence of the entropy of cyclobutane and cyclohexane. We believe we show that the concept of continuity in geometric symmetry provides a more natural and practical approach to the inherent link between symmetry and entropy, compared with the classical approach.

Localization for the Anderson Model on Trees with Finite Dimensions

Abstract.We introduce a family of trees that interpolate between the Bethe lattice and

Right Limits and Reflectionless Measures for CMV Matrices

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Large Deviations and the Lukic Conjecture

. We prove complete localization for the Anderson model on any member of that family.