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Barry Simon

Publications and source records attributed to Barry Simon.

At least 19 recordsLinked to original sources
arXiv2023

A Tale of Three Coauthors

We tell the story of the discovery of an interesting bound on finite sums and its application to comparison of Ising models.

math-ph↗
arXiv2023

A Useful Formula for Periodic Jacobi Matrices on Trees

We introduce a function of the density of states for periodic Jacobi matrices on trees and prove a useful formula for it. This allows new, streamlined proofs of the gap labeling and Aomoto index theorems. We prove a version of this new formula for the Anderson model on trees.

math.SP↗
arXiv2023

Remembrances of Derek William Robinson (25 June 1935 - 31 August 2021)

This arXiv submission is posted primarily to make available some additional material to what we prepared for the Derek Memorial article published in the Notices of the AMS. We start (with the permission of the AMS) with the text of our submission close to the final published version (which you can get online. After that we include some autobiographical notes that Derek dictated to Louisa Barnsley (whom we thank) during his final illness. Finally some unpublished notes Derek wrote on the history of his seminal monographs with Bratelli. We thank Marion Robinson for permission and encouragement to post these items.

math.HO↗
arXiv2023

Fritz and Me-Perspectives in Inverse Spectral Theory

This is a pedagogic introduction to certain aspects of inverse spectral theory for Schr\"{o}dinger operators and Jacobi matrices that revolves around my joint work with Fritz Gesztesy whose $70^{th}$ birthday we are honoring.

math.SP↗
arXiv2022

Comparison of Ising Models Under Change of Apriori Measure

We study comparison of correlation functions for ferromagnetic generalized Ising models with two different apriori measures. One purpose of this note is to publicize some unpublished 45 year old work of Daniel Wells on the issue. We then prove results for the apriori measures associated to one component of D-vectors uniformly distributed on the unit sphere and also the case of spin S (2S +1 equally spaced values symmetric about 0 and with equal weights) that improves some 50 year old bounds of Griffiths on transition temperatures.

math-ph↗
arXiv2021

Widom Factors and Szeg\H{o}-Widom Asymptotics, a Review

We survey results on Chebyshev polynomials centered around the work of H. Widom. In particular, we discuss asymptotics of the polynomials and their norms and general upper and lower bounds for the norms. Several open problems are also presented.

math.CA↗
arXiv2020

Twelve Tales in Mathematical Physics: An Expanded Heinemann Prize Lecture

This is an extended version of my 2018 Heinemann prize lecture describing the work for which I got the prize. The citation is very broad so this describes virtually all my work prior to 1995 and some afterwards. It discusses work in non-relativistic quantum mechanics, constructive quantum field theory and statistical mechanics.

math-ph↗
arXiv2020

Periodic Boundary Conditions for Periodic Jacobi Matrices on Trees

We consider matrices on infinite trees which are universal covers of Jacobi matrices on finite graphs. We are interested in the question of the existence of sequences of finite covers whose normalized eigenvalue counting measures converge to the density of states of the operator on the infinite tree. We first of all construct a simple example where this convergence fails and then discuss two ways of constructing the required sequences: with random boundary conditions and through normal subgroups.

math.SP↗
arXiv2020

Remarks on Periodic Jacobi Matrices on Trees

We look at periodic Jacobi matrices on trees. We provide upper and lower bounds on the gap of such operators analogous to the well known gap in the spectrum of the Laplacian on the upper half-plane with hyperbolic metric. We make some conjectures about antibound states and make an interesting observation for what [3] calls the rg-model.

math.SP↗
arXiv2020

Asymptotics of Chebyshev Polynomials, V. Residual Polynomials

We study residual polynomials, $R_{x_0,n}^{(\mathfrak{e})}$, $\mathfrak{e}\subset\mathbb{R}$, $x_0\in\mathbb{R}\setminus\mathfrak{e}$, which are the degree at most $n$ polynomials with $R(x_0)=1$ that minimize the $\sup$ norm on $\mathfrak{e}$. New are upper bounds on their norms (that are optimal in some cases) and Szeg\H{o}--Widom asymptotics under fairly general circumstances. We also discuss several illuminating examples and some results in the complex case.

math.CA↗
arXiv2019

Periodic Jacobi Matrices on Trees

We begin the systematic study of the spectral theory of periodic Jacobi matrices on trees including a formal definition. The most significant result that appears here for the first time is that these operators have no singular continuous spectrum. We review important previous results of Sunada and Aomoto and present several illuminating examples. We present many open problems and conjectures that we hope will stimulate further work.

math.SP↗
arXiv2019

Proof of the Strong Scott Conjecture for Chandrasekhar Atoms

We consider a large neutral atom of atomic number $Z$, taking relativistic effects into account by assuming the dispersion relation $\sqrt{c^2p^2+c^4}$. We study the behavior of the one-particle ground state density on the length scale $Z^{-1}$ in the limit $Z,c\to\infty$ keeping $Z/c$ fixed and find that the spherically averaged density as well as all individual angular momentum densities separately converge to the relativistic hydrogenic ones. This proves the generalization of the strong Scott conjecture for relativistic atoms and shows, in particular, that relativistic effects occur close to the nucleus. Along the way we prove upper bounds on the relativistic hydrogenic density.

math-ph↗