arXiv subjects

Jeffrey Schenker

Publications and source records attributed to Jeffrey Schenker.

31 records · Page 2Linked to original sources

arXiv2017

Localization in the Disordered Holstein model

The Holstein model describes the motion of a tight-binding tracer particle interacting with a field of quantum harmonic oscillators. We consider this model with an on-site random potential. Provided the hopping amplitude for the particle is small, we prove localization for matrix elements of the resolvent, in particle position and in the field Fock space. These bounds imply a form of dynamical localization for the particle position that leaves open the possibility of resonant tunneling in Fock space between equivalent field configurations.

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↗

arXiv2015

Quantum Brownian motion induced by thermal noise in the presence of disorder

The motion of a quantum particle hopping on a simple cubic lattice under the influence of thermal noise and of a static random potential is expected to be diffusive, i.e., the particle is expected to exhibit `quantum Brownian motion', no matter how weak the thermal noise is. This is shown to be true in a model where the dynamics of the particle is governed by a Lindblad equation for a one-particle density matrix. The generator appearing in this equation is the sum of two terms: a Liouvillian corresponding to a random Schrödinger operator and a Lindbladian describing the effect of thermal noise in the kinetic limit. Under suitable but rather general assumptions on the Lindbladian, the diffusion constant characterizing the asymptotics of the motion of the particle is proven to be strictly positive and finite. If the disorder in the random potential is so large that transport is completely suppressed in the limit where the thermal noise is turned off, then the diffusion constant tends to $0$ proportional to the coupling of the particle to the heat bath.

math-ph↗

arXiv2015

Diffusion in the Mean for an Ergodic Schrödinger Equation Perturbed by a Fluctuating Potential

Diffusive scaling of position moments and a central limit theorem are obtained for the mean position of a quantum particle hopping on a cubic lattice and subject to a random potential consisting of a large static part and a small part that fluctuates stochastically in time. If the static random potential is strong enough to induce complete localization in the absence of time dependent noise, then the diffusion constant is shown to go to zero proportional to the square of the strength of the time dependent part.

math-ph↗

arXiv2014

How large is large? Estimating the critical disorder for the Anderson model

Complete localization is shown to hold for the $d$-dimensional Anderson model with uniformly distributed random potentials provided the disorder strength $λ>λ_{And}$ where $λ_{\text{And}}$ satisfies $λ_{\text{And}}=μ_d e \ln λ_{\text{And}}$ with $μ_d$ the self-avoiding walk connective constant for the lattice $\mathbb{Z}^d$. Notably $λ_{\text{And}}$ is precisely the large disorder threshold proposed by Anderson in 1958.

math-ph↗

arXiv2014

Diffusive scaling for all moments of the Markov Anderson model

We consider a tight-binding Schroedinger equation with time dependent diagonal noise, given as a function of a Markov process. This model was considered previously by Kang and Schenker (J. Stat. Phys., 134(5-6):1005, arXiv:0808.2784), who proved that the wave propagates diffusively. We revisit the proof of diffusion so as to obtain a uniform bound on exponential moments of the wave amplitude and a central limit theorem that implies, in particular, diffusive scaling for all position moments of the mean wave amplitude.

math-ph↗

arXiv2010

Diffusive propagation of wave packets in a fluctuating periodic potential

We consider the evolution of a tight binding wave packet propagating in a fluctuating periodic potential. If the fluctuations stem from a stationary Markov process satisfying certain technical criteria, we show that the square amplitude of the wave packet after diffusive rescaling converges to a superposition of solutions of a heat equation.

math-ph↗

arXiv2010

Estimating complex eigenvalues of non-self-adjoint Schrödinger operators via complex dilations

The phenomenon "hypo-coercivity," i.e., the increased rate of contraction for a semi-group upon adding a large skew-adjoint part to the generator, is considered for 1D semigroups generated by the Schrödinger operators $-\partial^2_x + x^2 + iγ f (x)$ with a complex potential. For $f$ of the special form$ f (x) = 1/(1 + |x|^κ)$, it is shown using complex dilations that the real part of eigenvalues of the operator are larger than a constant times $|γ|^{2/(κ+2)}$.

math-ph↗

arXiv2009

Diffusion of wave packets in a Markov random potential

We consider the evolution of a tight binding wave packet propagating in a time dependent potential. If the potential evolves according to a stationary Markov process, we show that the square amplitude of the wave packet converges, after diffusive rescaling, to a solution of a heat equation.

math-ph↗

arXiv2009

Eigenvector localization for random band matrices with power law band width

It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors, provided the band width $W$ raised to a power $μ$ remains smaller than the matrix size $N$. For a Gaussian band ensemble, with matrix elements given by i.i.d. centered Gaussians within a band of width $W$, the estimate $μ\le 8$ holds.

math-ph↗

arXiv2007

Hamiltonian extension and eigenfunctions for a time dispersive dissipative string

We carry out a detailed analysis of a time dispersive dissipative (TDD) string, using our recently developed conservative and Hamiltonian extensions of TDD systems. This analysis of the TDD string includes, in particular: (i) an explicit construction of its conservative Hamiltonian extension, consisting of the physical string coupled to "hidden strings;" (ii) explicit formulas for energy and momentum densities in the extended system, providing a transparent physical picture accounting precisely for the dispersion and dissipation; (iii) the eigenmodes for the extended string system, which provide an eigenmode expansion for solutions to the TDD wave equation governing the TDD string. In particular, we find that in an eigenmode for the extended system the displacement of the physical string does not satisfy the formal eigenvalue problem, but rather an equation with a source term resulting from the excitation of the hidden strings. The obtained results provide a solid basis for the rigorous treatment of the long standing problem of scattering by a TDD scatterer, illustrated here by the computation of scattering states for a string with dissipation restricted to a half line.

math-ph↗

arXiv2006

Gaussian fluctuations for random matrices with correlated entries

For random matrix ensembles with non-gaussian matrix elements that may exhibit some correlations, it is shown that centered traces of polynomials in the matrix converge in distribution to a Gaussian process whose covariance matrix is diagonal in the basis of Chebyshev polynomials. The proof is combinatorial and adapts Wigner's argument showing the convergence of the density of states to the semicircle law.

math-ph↗