arXiv subjects
Publications and source records attributed to Barry Simon.
We determine which sets saturate the Szeg}o and Schiefermayr lower bounds on the norms of Chebyshev Polynomials. We also discuss sets that saturate the Totik--Widom upper bound.
Given two orthogonal projections P and Q, we are interested in all unitary operators U such that UP=QU and UQ=PU. Such unitaries U have previously been constructed by Wang, Du, and Dou and also by one of the authors. One purpose of this note is to compare these constructions. Very recently, Dou, Shi, Cui, and Du described all unitaries U with the required property. Their proof is via the two projections theorem by Halmos. We here give a proof based on the supersymmetric approach by Avron, Seiler, and one of the authors.
We review the work of Tosio Kato on the mathematics of non--relativistic quantum mechanics and some of the research that was motivated by this. Topics include analytic and asymptotic eigenvalue perturbation theory, Temple--Kato inequality, self--adjointness results, quadratic forms including monotone convergence theorems, absence of embedded eigenvalues, trace class scattering, Kato smoothness, the quantum adiabatic theorem and Kato's ultimate Trotter Product Formula.
Based at a talk given at the Kato Centennial Symposium in Sept. 2017, this article discusses the scientific life and some of the scientific work of T. Kato.
Let $P$ and $Q$ be two orthogonal projections on a separable Hilbert space, $\calH$. Wang, Du and Dou proved that there exists a unitary, $U$, with $UPU^{-1} =Q, \quad UQU^{-1} = P$ if and only if $\dim(\ker P \cap \ker(1-Q)) = \dim(\ker Q \cap \ker(1-P))$ (both may be infinite). We provide a new proof using the supersymmetric machinery of Avron, Seiler and Simon.
This is a pedagogical exposition of the large deviation approach to sum rules pioneered by Gamboa, Nagel and Rouault. We'll explain how to use their ideas to recover the Szeg}o and Killip{ Simon Theorems. The primary audience is spectral theorists and people working on orthogonal polynomials who have limited familiarity with the theory of large deviations.
Short blurb for invited talk at AMS annual meeting in Atlanta; will appear in January AMS Notices
We consider a gas of fermions at zero temperature and low density, interacting via a microscopic two body potential which admits a bound state. The particles are confined to a domain with Dirichlet (i.e. zero) boundary conditions. Starting from the microscopic BCS theory, we derive an effective macroscopic Gross-Pitaevskii (GP) theory describing the condensate of fermion pairs. The GP theory also has Dirichlet boundary conditions. Along the way, we prove that the GP energy, defined with Dirichlet boundary conditions on a bounded Lipschitz domain, is continuous under interior and exterior approximations of that domain.
Laptev and Safronov conjectured that any non-positive eigenvalue of a Schrödinger operator $-Δ+V$ in $L^2(\mathbb R^ν)$ with complex potential has absolute value at most a constant times $\|V\|_{γ+ν/2}^{(γ+ν/2)/γ}$ for $0<γ\leqν/2$ in dimension $ν\geq 2$. We prove this conjecture for radial potentials if $0<γ<ν/2$ and we `almost disprove' it for general potentials if $1/2<γ<ν/2$. In addition, we prove various bounds that hold, in particular, for positive eigenvalues.
We consider Chebyshev polynomials, $T_n(z)$, for infinite, compact sets $\frak{e} \subset \mathbb{R}$ (that is, the monic polynomials minimizing the sup-norm, $\Vert T_n \Vert_{\frak{e}}$, on $\frak{e}$). We resolve a $45+$ year old conjecture of Widom that for finite gap subsets of $\mathbb{R}$, his conjectured asymptotics (which we call Szegő-Widom asymptotics) holds. We also prove the first upper bounds of the form $\Vert T_n \Vert_{\frak{e}} \leq Q C({\frak{e}})^n$ (where $C(\frak{e})$ is the logarithmic capacity of $\frak{e}$) for a class of $\frak{e}$'s with an infinite number of components, explicitly for those $\frak{e} \subset \mathbb{R}$ that obey a Parreau-Widom condition.
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 $μ(\{x\})=0$.
Let $\fre\subset\bbR$ be a finite union of $\ell+1$ disjoint closed intervals and denote by $ω_j$ the harmonic measure of the $j$ leftmost bands. The frequency module for $\fre$ is the set of all integral combinations of $ω_1,..., ω_\ell$. Let $\{\tilde{a}_n, \tilde{b}_n\}_{n=1}^\infty$ be a point in the isospectral torus for $\fre$ and $\tilde{p}_n$ its orthogonal polynomials. Let $\{a_n,b_n\}_{n=1}^\infty$ be a half-line Jacobi matrix with $a_n = \tilde{a}_n + δa_n$, $b_n = \tilde{b}_n + δb_n$. Suppose \[ \sum_{n=1}^\infty %(\abs{a_n-\tilde{a}_n}^2 + \abs{b_n-\tilde{b}_n}^2) <\infty \abs{δa_n}^2 + \abs{δb_n}^2 <\infty \] and $\sum_{n=1}^N e^{2πiωn} δa_n$, $\sum_{n=1}^N e^{2πiωn} δb_n$ have finite limits as $N\to\infty$ for all $ω$ in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to $ω$, then for $z\in\bbC\setminus\bbR$, $p_n(z)/\tilde{p}_n(z)$ has a limit as $n\to\infty$. Moreover, we show that there are non-Szegő class $J$'s for which this holds.
We show that for many families of OPUC, one has $||φ'_n||_2/n -> 1$, a condition we call normal behavior. We prove that this implies $|α_n| -> 0$ and that it holds if the sequence $α_n$ is in $\ell^1$. We also prove it is true for many sparse sequences. On the other hand, it is often destroyed by the insertion of a mass point.
We present and exploit an analogy between lack of absolutely continuous spectrum for Schroedinger operators and natural boundaries for power series. Among our new results are generalizations of Hecke's example and natural boundary examples for random power series where independence is not assumed.
We prove bounds of the form $\sum_{e\in I\capσ_\di (H)} \dist (e,σ_\e (H))^{1/2} \leq L^1$-norm of a perturbation, where $I$ is a gap. Included are gaps in continuum one-dimensional periodic Schrödinger operators and finite gap Jacobi matrices where we get a generalized Nevai conjecture about an $L^1$ condition implying a Szegő condition. One key is a general new form of the Birman--Schwinger bound in gaps.
Let $\fre\subset\bbR$ be a finite union of disjoint closed intervals. We study measures whose essential support is $\fre$ and whose discrete eigenvalues obey a 1/2-power condition. We show that a Szegő condition is equivalent to \[ \limsup \f{a_1... a_n}{\ca(\fre)^n}>0 \] (this includes prior results of Widom and Peherstorfer--Yuditskii). Using Remling's extension of the Denisov--Rakhmanov theorem and an analysis of Jost functions, we provide a new proof of Szegő asymptotics, including $L^2$ asymptotics on the spectrum. We use heavily the covering map formalism of Sodin--Yuditskii as presented in our first paper in this series.
Let $\fre$ be a homogeneous subset of $\bbR$ in the sense of Carleson. Let $μ$ be a finite positive measure on $\bbR$ and $H_μ(x)$ its Hilbert transform. We prove that if $\lim_{t\to\infty} t \abs{\fre\cap\{x\mid\abs{H_μ(x)}>t\}}=0$, then $μ_s(\fre)=0$, where $μ_\s$ is the singular part of $μ$.