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Mira Shamis

Publications and source records attributed to Mira Shamis.

15 recordsLinked to original sources
arXiv2025

Area Law for the entanglement entropy of free fermions in nonrandom ergodic field

This paper deals with the asymptotic behaviour of a widely used correlation characteristic in large quantum systems. The correlations are known as quantum entanglement, the characteristic is called entanglement entropy, and the system is an ideal gas of spinless lattice fermions. The system is determined by its one-body Hamiltonian. It is shown in EPS [18] that if the Hamiltonian is an ergodic finite difference operator with an exponentially decaying spectral projection, then the asymptotic form of the entanglement entropy is the so-called area law. However, the only class of one-body Hamiltonians for which this spectral condition was verified is that consisting of discrete Schr\"odinger operators with random potential. In this paper, we prove the validity of the area law for several classes of Schr\"odinger operators whose potentials are ergodic but not random. We begin with quasiperiodic and limit-periodic operators and then move on to the interesting and highly non-trivial case of potentials generated by subshifts of finite type. These arose in the theory of dynamical systems when studying non-random chaotic phenomena. The corresponding asymptotic study requires quite an involved spectral analysis. Consequently, the majority of the paper is devoted to the proof and application of a variety of spectral properties of the operators in question, in particular we prove uniform localisation of the ejgenfunctions for the Maryland model and the exponential decay of the eihgenfunction correlator for a variety of models . We believe that these properties are of considerable independent interest.

math-ph↗
arXiv2023

On the abominable properties of the almost Mathieu operator with well approximated frequencies

We show that some spectral properties of the almost Mathieu operator with frequency well approximated by rationals can be as poor as at all possible in the class of all one-dimensional discrete Schroedinger operators. For the class of critical coupling, we show that the Hausdorff measure of the spectrum may vanish (for appropriately chosen frequencies) whenever the gauge function tends to zero faster than logarithmically. For arbitrary coupling, we show that modulus of continuity of the integrated density of states can be arbitrary close to logarithmic; we also prove a similar result for the Lyapunov exponent as a function of the spectral parameter. Finally, we show that (for any coupling) there exist frequencies for which the spectrum is not homogeneous in the sense of Carleson, and, moreover, fails the Parreau-Widom condition. The frequencies for which these properties hold are explicitly described in terms of the growth of the denominators of the convergents.

math-ph↗
arXiv2021

Upper bounds on quantum dynamics in arbitrary dimension

Motivated by the research on upper bounds on the rate of quantum transport for one-dimensional operators, particularly, the recent works of Jitomirskaya--Liu and Jitomirskaya--Powell and the earlier ones of Damanik--Tcheremchantsev, we propose a method to prove similar bounds in arbitrary dimension. The method applies both to Schroedinger and to long-range operators. In the case of ergodic operators, one can use large deviation estimates for the Green function in finite volumes to verify the assumptions of our general theorem. Such estimates have been proved for numerous classes of quasiperiodic operators in one and higher dimension, starting from the works of Bourgain, Goldstein, and Schlag. One of the applications is a power-logarithmic bound on the quantum transport defined by a multidimensional discrete Schrödinger (or even long-range) operator associated with an irrational shift, valid for all Diophantine frequencies and uniformly for all phases. To the best of our knowledge, these are the first results on the quantum dynamics for quasiperiodic operators in dimension greater than one that do not require exclusion of a positive measure of phases. Moreover, and in contrast to localisation, the estimates are uniform in the phase. The arguments are also applicable to ergodic operators corresponding to other kinds of base dynamics, such as the skew-shift.

math-ph↗
arXiv2019

On the continuity of the integrated density of states in the disorder

Recently, Hislop and Marx studied the dependence of the integrated density of states on the underlying probability distribution for a class of discrete random Schrödinger operators, and established a quantitative form of continuity in weak* topology. We develop an alternative approach to the problem, based on Ky Fan inequalities, and establish a sharp version of the estimate of Hislop and Marx. We also consider a corresponding problem for continual random Schrödinger operators on $\R^d$.

math-ph↗
arXiv2017

Matrix regularizing effects of Gaussian perturbations

The addition of noise has a regularizing effect on Hermitian matrices. This effect is studied here for $H=A+V$, where $A$ is the base matrix and $V$ is sampled from the GOE or the GUE random matrix ensembles. We bound the mean number of eigenvalues of $H$ in an interval, and present tail bounds for the distribution of the Frobenius and operator norms of $H^{-1}$ and for the distribution of the norm of $H^{-1}$ applied to a fixed vector. The bounds are uniform in $A$ and exceed the actual suprema by no more than multiplicative constants. The probability of multiple eigenvalues in an interval is also estimated.

math.PR↗
arXiv2016

On the Wegner orbital model

The Wegner orbital model is a class of random operators introduced by Wegner to model the motion of a quantum particle with many internal degrees of freedom (orbitals) in a disordered medium. We consider the case when the matrix potential is Gaussian, and prove three results: localisation at strong disorder, a Wegner-type estimate on the mean density of eigenvalues, and a Minami-type estimate on the probability of having multiple eigenvalues in a short interval. The last two results are proved in the more general setting of deformed block-Gaussian matrices, which includes a class of Gaussian band matrices as a special case. Emphasis is placed on the dependence of the bounds on the number of orbitals. As an additional application, we improve the upper bound on the localisation length for one-dimensional Gaussian band matrices.

math-ph↗
arXiv2014

Resonances and Partial Delocalization on the Complete Graph

Random operators may acquire extended states formed from a multitude of mutually resonating local quasi-modes. This mechanics is explored here in the context of the random Schrödinger operator on the complete graph. The operators exhibits local quasi modes mixed through a single channel. While most of its spectrum consists of localized eigenfunctions, under appropriate conditions it includes also bands of states which are delocalized in the $\ell^1$-though not in $\ell^2$-sense, where the eigenvalues have the statistics of Šeba spectra. The analysis proceeds through some general observations on the scaling limits of random functions in the Herglotz-Pick class. The results are in agreement with a heuristic condition for the emergence of resonant delocalization, which is stated in terms of the tunneling amplitude among quasi-modes.

math-ph↗
arXiv2014

Bounds on the Lyapunov exponent via crude estimates on the density of states

We study the Chirikov (standard) map at large coupling $λ\gg 1$, and prove that the Lyapounov exponent of the associated Schroedinger operator is of order $\log λ$ except for a set of energies of measure $\exp(-c λ^β)$ for some $1 < β< 2$. We also prove a similar (sharp) lower bound on the Lyapunov exponent (outside a small exceptional set of energies) for a large family of ergodic Schroedinger operators, the prime example being the $d$-dimensional skew shift.

math-ph↗
arXiv2013

Density of states for GUE, GOE, and interpolating ensembles through supersymmetric approach

We use the supersymmetric formalism to derive an integral formula for the density of states of the Gaussian Orthogonal Ensemble, and then apply saddle-point analysis to give a new derivation of the 1/N-correction to Wigner's law. This extends the work of Disertori on the Gaussian Unitary Ensemble. We also apply our method to the interpolating ensembles of Mehta--Pandey.

math-ph↗
arXiv2013

Resonant delocalization on the Bethe strip

Recently, Aizenman and Warzel discovered a mechanism for the appearance of absolutely continuous spectrum for random Schroedinger operators on the Bethe lattice through rare resonances (resonant delocalization). We extend their analysis to operators with matrix-valued random potentials drawn from ensembles such as the Gaussian Orthogonal Ensemble. These operators can be viewed as random operators on the Bethe strip, a graph (lattice) with loops.

math-ph↗
arXiv2012

Localisation for non-monotone Schroedinger operators

We study localisation effects of strong disorder on the spectral and dynamical properties of (matrix and scalar) Schroedinger operators with non-monotone random potentials, on the d-dimensional lattice. Our results include dynamical localisation, i.e. exponentially decaying bounds on the transition amplitude in the mean. They are derived through the study of fractional moments of the resolvent, which are finite due to resonance-diffusing effects of the disorder. One of the byproducts of the analysis is a nearly optimal Wegner estimate. A particular example of the class of systems covered by our results is the discrete alloy-type Anderson model.

math-ph↗
arXiv2011

Some connections between almost periodic and periodic discrete Schroedinger operators with analytic potentials

We study discrete Schroedinger operators with analytic potentials. In particular, we are interested in the connection between the absolutely continuous spectrum in the almost periodic case and the spectra in the periodic case. We prove a weak form of a precise conjecture relating the two. We also bound the measure of the spectrum in the periodic case in terms of the Lyapunov exponent in the almost periodic case. In the proofs, we use a partial generalization of Chambers' formula. As an additional application of this generalization, we provide a new proof of Herman's lower bound for the Lyapunov exponent.

math.SP↗
arXiv2010

On the Measure of the Absolutely Continuous Spectrum for Jacobi Matrices

We apply the methods of classical approximation theory (extreme properties of polynomials) to study the essential support $Σ_{ac}$ of the absolutely continuous spectrum of Jacobi matrices. First, we prove an upper bound on the measure of $Σ_{ac}$ which takes into account the value distribution of the diagonal elements, and implies the bound due to Deift-Simon and Poltoratski-Remling. Second, we generalise the differential inequality of Deift-Simon for the integrated density of states associated with the absolutely continuous spectrum to general Jacobi matrices.

math.SP↗