arXiv subjects

Uzy Smilansky

Publications and source records attributed to Uzy Smilansky.

At least 19 recordsLinked to original sources

arXiv2026

Characterization of Chaotic Evolution in Quantum Systems Induced by Random Hermitian Matrices

In a recent paper, a semiclassical Lyapunov exponent associated with a quantum Hamiltonian represented by a finite-dimensional Hermitian matrix was defined and placed on a mathematical foundation. The Lyapunov exponent characterizes the early stages of the evolution toward the ergodic state, while the late stages are characterized by the spectral gap of the corresponding Markov matrix. Here, we apply this formalism to five random-matrix ensembles. For each ensemble, we derive the mean Lyapunov exponent, its variance, and the spectral gap as functions of energy. We also present the corresponding thermal averages. Extensive numerical data are compared with the theoretical predictions.

nlin.CD↗

arXiv2024

Information scrambling and chaos induced by a Hermitian Matrix

Given an arbitrary $V \times V$ Hermitian matrix, considered as a finite discrete quantum Hamiltonian, we use methods from graph and ergodic theories to construct a \textit{quantum Poincar\'e map} at energy $E$ and a corresponding stochastic \textit{classical Poincar\'e-Markov map} at the same energy on an appropriate discrete \textit{phase space}. This phase space consists of the directed edges of a graph with $V$ vertices that are in one-to-one correspondence with the non-vanishing off-diagonal elements of $H$. The correspondence between quantum Poincar\'e map and classical Poincar\'e-Markov map is an alternative to the standard quantum-classical correspondence based on a classical limit $\hbar \to 0$. Most importantly it can be constructed where no such limit exists. Using standard methods from ergodic theory we then proceed to define an expression for the \textit{Lyapunov exponent} $\Lambda(E)$ of the classical map. It measures the rate of loss of classical information in the dynamics and relates it to the separation of stochastic \textit{classical trajectories} in the phase space. We suggest that loss of information in the underlying classical dynamics is an indicator for quantum information scrambling.

quant-ph↗

arXiv2022

Time evolution and the Schr\"odinger equation on time dependent quantum graphs

The purpose of the present paper is to discuss the time dependent Schr\"odinger equation on a metric graph with time-dependent edge lengths, and the proper way to pose the problem so that the corresponding time evolution is unitary. We show that the well posedness of the Schr\"odinger equation can be guaranteed by replacing the standard Kirchhoff Laplacian with a magnetic Schr\"odinger operator with a harmonic potential. We then generalize the result to time dependent families of vertex conditions. We also apply the theory to show the existence of a geometric phase associated with a slowly changing quantum graph.

math-ph↗

arXiv2022

The Kronig-Penney model in a quadratic channel with $δ$ interactions. II : Scattering approach

The main purpose of the present paper is to introduce a scattering approach to the study of the Kronig-Penney model in a quadratic channel with $δ$ interactions, which was discussed in full generality in the first paper of the present series. In particular, a secular equation whose zeros determine the spectrum will be written in terms of the scattering matrix from a single $δ$. The advantages of this approach will be demonstrated in addressing the domain with total energy $E\in [0,\frac{1}{2})$, namely, the energy interval where, for under critical interaction strength, a discrete spectrum is known to exist for the single $δ$ case. Extending this to the study of the periodic case reveals quite surprising behavior of the Floquet spectra and the corresponding spectral bands. The computation of these bands can be carried out numerically, and the main features can be qualitatively explained in terms of a semi-classical framework which is developed for the purpose.

math-ph↗

arXiv2022

A trace formula for metric graphs with piecewise constant potentials and multi-mode graphs

We generalize the scattering approach to quantum graphs to quantum graphs with with piecewise constant potentials and multiple excitation modes. The free single-mode case is well-known and leads to the trace formulas of Roth, Kottos and Smilansky. By introducing an effective reduced scattering picture we are able to introduce new exact trace formulas in the more general setting. The latter are derived and discussed in details with some numerical examples for illustration. Our generalization is motivated by both experimental applications and fundamental theoretical considerations. The free single-mode quantum graphs are an extreme idealization of reality that, due to the simplicity of the model allows to understand a large number of generic or universal phenomena. We lift some of this idealization by considering the influence of evanescent modes that only open above threshold energies. How to do this theoretically in a closed model in general is a challenging question of fundamental theoretical interest and we achieve this here for quantum graphs.

quant-ph↗

arXiv2021

Can one hear a matrix? Recovering a real symmetric matrix from its spectral data

The spectrum of a real and symmetric $N\times N$ matrix determines the matrix up to unitary equivalence. More spectral data is needed together with some sign indicators to remove the unitary ambiguities. In the first part of this work we specify the spectral and sign information required for a unique reconstruction of general matrices. More specifically, the spectral information consists of the spectra of the $N$ nested main minors of the original matrix of the sizes $1,2,\dots,N$. However, due to the complicated nature of the required sign data, improvements are needed in order to make the reconstruction procedure feasible. With this in mind, the second part is restricted to banded matrices where the amount of spectral data exceeds the number of the unknown matrix entries. It is shown that one can take advantage of this redundancy to guarantee unique reconstruction of {\it generic} matrices, in other words, this subset of matrices is open, dense and of full measure in the set of real, symmetric and banded matrices. It is shown that one can optimize the ratio between redundancy and genericity by using the freedom of choice of the spectral information input. We demonstrate our constructions in detail for pentadiagonal matrices.

math-ph↗

arXiv2020

The Statistics of Spectral Shifts due to Finite Rank Perturbations

This article is dedicated to the following class of problems. Start with an $N\times N$ Hermitian matrix randomly picked from a matrix ensemble - the reference matrix. Applying a rank-$t$ perturbation to it, with $t$ taking the values $1\le t \le N$, we study the difference between the spectra of the perturbed and the reference matrices as a function of $t$ and its dependence on the underlying universality class of the random matrix ensemble. We consider both, the weaker kind of perturbation which either permutes or randomizes $t$ diagonal elements and a stronger perturbation randomizing successively $t$ rows and columns. In the first case we derive universal expressions in the scaled parameter $τ=t/N$ for the expectation of the variance of the spectral shift functions, choosing as random-matrix ensembles Dyson's three Gaussian ensembles. In the second case we find an additional dependence on the matrix size $N$.

cond-mat.stat-mech↗

arXiv2019

Trace formulas for general Hermitian matrices: Unitary scattering approach and periodic orbits on an associated graph

Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix $H$ to a unitary analogue. In the first type the unitary matrix is $e^{i(λ\II - H)}$ where $λ$ is the spectral parameter. The new feature is that the spectral parameter appears in the final form as an argument of Eulerian polynomials -- thus connecting the periodic orbits to combinatorial objects in a novel way. To obtain the second type, one expresses the input in terms of a unitary scattering matrix in a larger Hilbert space. One of the surprising features here is that the locations and radii of the spectral discs of Gershgorin's theorem appear naturally as the pole parameters of the scattering matrix. Both formulas are discussed and possible applications are outlined.

math-ph↗

arXiv2019

The Distribution of Path Lengths On Directed Weighted Graphs

We consider directed weighted graphs and define various families of path counting functions. Our main results are explicit formulas for the main term of the asymptotic growth rate of these counting functions, under some irrationality assumptions on the lengths of all closed orbits on the graph. In addition we assign transition probabilities to such graphs and compute statistics of the corresponding random walks. Some examples and applications are reviewed.

math.CO↗

arXiv2018

Delay-time distribution in the scattering of time-narrow wave packets (II) - Quantum Graphs

We apply the framework developed in the preceding paper in this series (Smilansky 2017 J. Phys. A: Math. Theor. 50, 215301) to compute the time-delay distribution in the scattering of ultra short radio frequency pulses on complex networks of transmission lines which are modeled by metric (quantum) graphs. We consider wave packets which are centered at high wave number and comprise many energy levels. In the limit of pulses of very short duration we compute upper and lower bounds to the actual time-delay distribution of the radiation emerging from the network using a simplified problem where time is replaced by the discrete count of vertex-scattering events. The classical limit of the time-delay distribution is also discussed and we show that for finite networks it decays exponentially, with a decay constant which depends on the graph connectivity and the distribution of its edge lengths. We illustrate and apply our theory to a simple model graph where an algebraic decay of the quantum time-delay distribution is established.

math-ph↗

arXiv2017

Edge switching transformations of quantum graphs

Discussed here are the effects of basics graph transformations on the spectra of associated quantum graphs. In particular it is shown that under an edge switch the spectrum of the transformed Schrödinger operator is interlaced with that of the original one. By implication, under edge swap the spectra before and after the transformation, denoted by $\{ E_n\}_{n=1}^{\infty}$ and $\{\widetilde E_n\}_{n=1}^{\infty}$ correspondingly, are level-2 interlaced, so that $E_{n-2}\le \widetilde E_n\le E_{n+2}$. The proofs are guided by considerations of the quantum graphs' discrete analogs.

math-ph↗

arXiv2017

A random walk approach to linear statistics in random tournament ensembles

We investigate the linear statistics of random matrices with purely imaginary Bernoulli entries of the form $H_{pq} = \overline{H}_{qp} = \pm i$, that are either independently distributed or exhibit global correlations imposed by the condition $\sum_{q} H_{pq} = 0$. These are related to ensembles of so-called random tournaments and random regular tournaments respectively. Specifically, we construct a random walk within the space of matrices and show that the induced motion of the first $k$ traces in a Chebyshev basis converges to a suitable Ornstein-Uhlenbeck process. Coupling this with Stein's method allows us to compute the rate of convergence to a Gaussian distribution in the limit of large matrix dimension.

math.PR↗

arXiv2017

Spectral statistics of the uni-modular ensemble

We investigate the spectral statistics of Hermitian matrices in which the elements are chosen uniformly from U (1), called the uni-modular ensemble (UME), in the limit of large matrix size. Using three complimentary methods; a supersymmetric integration method, a combinatorial graph-theoretical analysis and a Brownian motion approach, we are able to derive expressions for 1/N corrections to the mean spectral moments and also analyse the fluctuations about this mean. By addressing the same ensemble from three different point of view, we can critically compare their relative advantages and derive some new results.

math-ph↗

arXiv2017

Delay-time distribution in the scattering of time-narrow wave packets. (I)

This is the first of two subsequent publications where the probability distribution of delay-times in scattering of wave packets is discussed. The probability distribution is expressed in terms of the on-shell scattering matrix, the dispersion relation of the scattered beam and the wave packet envelope. In the monochromatic limit (poor time resolution) the mean delay-time coincides with the expression derived by Eisenbud and Wigner and generalized by Smith more than half a century ago. In the opposite limit, and within the semi-classical approximation, the resulting distribution coincides with the result obtained using classical mechanics or geometrical optics. The general expression interpolates smoothly between the two extremes. An application for the scattering of electromagnetic waves in networks of RF transmission lines will be discussed in the next paper to illustrate the method in an experimentally relevant context.

math-ph↗

arXiv2015

The probability distribution of spectral moments for the Gaussian beta-ensembles

We derive the joint probability distribution of the first two spectral moments for the G$β$E random matrix ensembles in N dimensions for any N. This is achieved by making use of two complementary invariants of the domain in $\mathbb{R}^N$ where the spectral moments are defined. Our approach is significantly different from those employed previously to answer related questions and potentially offers new insights. We also discuss the problems faced when attempting to include higher spectral moments.

math-ph↗

arXiv2015

On the Spectrum of Random Anti-symmetric and Tournament Matrices

We consider a discrete, non-Hermitian random matrix model, which can be expressed as a shift of a rank-one perturbation of an anti-symmetric matrix. We show that, asymptotically almost surely, the real parts of the eigenvalues of the non-Hermitian matrix around any fixed index remain interlaced with those of the anti-symmetric matrix. Along the way, we show that some tools recently developed to study the eigenvalue distributions of Hermitian matrices extend to the anti-symmetric setting.

math.PR↗

arXiv2015

Dyson's Brownian-motion model for random matrix theory - revisited. With an Appendix by Don Zagier

We offer an alternative viewpoint on Dyson's original paper regarding the application of Brownian motion to random matrix theory (RMT). In particular we show how one may use the same approach in order to study the stochastic motion in the space of matrix traces $t_n = \sum_{ν=1}^{N} λ_ν^n$, rather than the eigenvalues $λ_ν$. In complete analogy with Dyson we obtain a Fokker-Planck equation that exhibits a stationary solution corresponding to the joint probability density function in the space $t = (t_1,\ldots,t_n)$, which can in turn be related to the eigenvalues $λ= (λ_1,\ldots,λ_N)$. As a consequence two interesting combinatorial identities emerge, which are proved algebraically in the appendix. We also offer a number of comments on this version of Dyson's theory and discuss its potential advantages.

math-ph↗

arXiv2015

Spectral statistics of Bernoulli matrix ensembles - a random walk approach (I)

We investigate the eigenvalue statistics of random Bernoulli matrices, where the matrix elements are chosen independently from a binary set with equal probability. This is achieved by initiating a discrete random walk process over the space of matrices and analysing the induced random motion of the eigenvalues - an approach which is similar to Dyson's Brownian motion model but with important modifications. In particular, we show our process is described by a Fokker-Planck equation, up to an error margin which vanishes in the limit of large matrix dimension. The stationary solution of which corresponds to the joint probability density function of certain well-known fixed trace Gaussian ensembles.

math-ph↗