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Yoram Last

Publications and source records attributed to Yoram Last.

17 recordsLinked to original sources
arXiv2026

A Note on Extended States on the Bethe Lattice

We give a new proof of the existence of absolutely continuous spectrum for the weakly disordered Anderson model on the Bethe lattice. The argument follows a general cyclicity criterion for Anderson-type Hamiltonians and reduces the problem to showing that the spectral measures of two independent copies of the rooted tree are not mutually singular. This is detected by their Hellinger affinity, and the required weak-disorder estimate follows from a short harmonic and compactness argument. We also formulate an analogous quantitative cyclicity criterion for the Anderson model on $\mathbb{Z}^d$, expressed through the Schur complement of the Poisson transform of a $2\times2$ matrix spectral measure.

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↗
arXiv2017

$\ell^2$ bounded variation and absolutely continuous spectrum of Jacobi matrices

We disprove a conjecture of Breuer-Last-Simon concerning the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an $\ell^2$ bounded variation condition with step $q$. We prove existence of a.c. spectrum on a smaller set than that specified by the conjecture and prove that our result is optimal.

math.SP↗
arXiv2015

Conductance and absolutely continuous spectrum of 1D samples

We characterize the absolutely continuous spectrum of the one-dimensional Schrödinger operators $h=-Δ+v$ acting on $\ell^2(\mathbb{Z}_+)$ in terms of the limiting behavior of the Landauer-Büttiker and Thouless conductances of the associated finite samples. The finite sample is defined by restricting $h$ to a finite interval $[1,L]\cap\mathbb{Z}_+$ and the conductance refers to the charge current across the sample in the open quantum system obtained by attaching independent electronic reservoirs to the sample ends. Our main result is that the conductances associated to an energy interval $I$ are non-vanishing in the limit $L\to\infty$ iff ${\rm sp}_{\rm ac}(h)\cap I=\emptyset$. We also discuss the relationship between this result and the Schrödinger Conjecture.

math-ph↗
arXiv2014

Landauer-Büttiker and Thouless conductance

In the independent electron approximation, the average (energy/charge/entropy) current flowing through a finite sample S connected to two electronic reservoirs can be computed by scattering theoretic arguments which lead to the famous Landauer-Büttiker formula. Another well known formula has been proposed by Thouless on the basis of a scaling argument. The Thouless formula relates the conductance of the sample to the width of the spectral bands of the infinite crystal obtained by periodic juxtaposition of S. In this spirit, we define Landauer-Büttiker crystalline currents by extending the Landauer-Büttiker formula to a setup where the sample S is replaced by a periodic structure whose unit cell is S. We argue that these crystalline currents are closely related to the Thouless currents. For example, the crystalline heat current is bounded above by the Thouless heat current, and this bound saturates iff the coupling between the reservoirs and the sample is reflectionless. Our analysis leads to a rigorous derivation of the Thouless formula from the first principles of quantum statistical mechanics.

math.SP↗
arXiv2013

Stability of Asymptotics of Christoffel-Darboux Kernels

We study the stability of convergence of the Christoffel-Darboux kernel, associated with a compactly supported measure, to the sine kernel, under perturbations of the Jacobi coefficients of the measure. We prove stability under variations of the boundary conditions and stability in a weak sense under $\ell^1$ and random $\ell^2$ diagonal perturbations. We also show that convergence to the sine kernel at $x$ implies that $μ(\{x\})=0$.

math.SP↗
arXiv2008

Bulk Universality and Clock Spacing of Zeros for Ergodic Jacobi Matrices with A.C. Spectrum

By combining some ideas of Lubinsky with some soft analysis, we prove that universality and clock behavior of zeros for OPRL in the a.c. spectral region is implied by convergence of $\frac{1}{n} K_n(x,x)$ for the diagonal CD kernel and boundedness of the analog associated to second kind polynomials. We then show that these hypotheses are always valid for ergodic Jacobi matrices with a.c. spectrum and prove that the limit of $\frac{1}{n} K_n(x,x)$ is $ρ_\infty(x)/w(x)$ where $ρ_\infty$ is the density of zeros and $w$ is the a.c. weight of the spectral measure.

math.SP↗
arXiv2008

The Nevai Condition

We study Nevai's condition that for orthogonal polynomials on the real line, $K_n(x,x_0)^2 K_n(x_0,x_0)^{-1} dρ(x)\toδ_{x_0}$ where $K_n$ is the CD kernel. We prove that it holds for the Nevai class of a finite gap set uniformly on the spectrum and we provide an example of a regular measure on $[-2,2]$ where it fails on an interval.

math.SP↗
arXiv2007

Monotone Jacobi parameters and non-Szego weights

We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for $a_n\equiv 1$, $b_n =-C n^{-β}$ ($0<β< \frac23)$, one has $dμ(x)= w(x) dx$ on $(-2,2)$, and near $x=2$, $w(x)=e^{-2Q(x)}$ where \[ Q(x)=β^{-1} C^{\frac{1}β} \frac{Γ(\frac32)Γ(\frac{1}β}-\frac12)(2-x)^{\frac12 -\frac{1}β}}{Γ(\frac{1}β+1)}(1+O((2-x))) \]

math.SP↗
arXiv2006

Fine Structure of the Zeros of Orthogonal Polynomials, IV. A Priori Bounds and Clock Behavior

We prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions (Jacobi parameters approach the free ones and are of bounded variation). We prove that for ergodic discrete Schrodinger operators, Poisson behavior implies positive Lyapunov exponent. Both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs.

math.SP↗
arXiv2005

The essential spectrum of Schrodinger, Jacobi, and CMV operators

We provide a very general result that identifies the essential spectrum of broad classes of operators as exactly equal to the closure of the union of the spectra of suitable limits at infinity. Included is a new result on the essential spectra when potentials are asymptotic to isospectral tori. We also recover with a unified framework the HVZ theorem and Krein's results on orthogonal polynomials with finite essential spectra.

math.SP↗
arXiv1999

Solutions, Spectrum, and Dynamica for Schrödinger Operators on Infinite Domains

Let H be a Schrödinger operator defined on an unbounded domain D in R^d with Dirichlet boundary conditions (D may equal R^d in particular). Let u(x,E) be a solution of the Schrödinger equation (H-E)u(x,E)=0, and let B_R denote a ball of radius R centered at zero. We show relations between the rate of growth of the L^2 norm \|u(x,E)\|_{L^2(B_R \cap D)} of such solutions as R goes to infinity, and continuity properties of spectral measures of the operator H. These results naturally lead to new criteria for identification of various spectral properties. We also prove new fundamental relations berween the rate of growth of L^2 norms of generalized eigenfunctions, dimensional properties of the spectral measures, and dynamical properties of the corresponding quantum systems. We apply these results to study transport properties of some particular Schrödinger operators.

math.SP↗