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Barry Simon

Publications and source records attributed to Barry Simon.

At least 91 records · Page 5Linked to original sources
arXiv2004

Higher-Order Szego Theorems With Two Singular Points

We consider probability measures, $dμ=w(θ) \f{dθ}{2π} +dμ_\s$, on the unit circle, $\partial\bbD$, with Verblunsky coefficients, $\{α_j\}_{j=0}^\infty$. We prove for $θ_1\neqθ_2$ in $[0,2π)$ and $(δβ)_j=β_{j+1}$ that \[ \int [1-\cos(θ-θ_1)][1-\cos(θ-θ_2)] \log w(θ) \f{dθ}{2π} >-\infty \] if and only if \[ \sum_{j=0}^\infty \bigl|\bigl\{(δ-e^{-iθ_2}) (δ-e^{-iθ_1}) α\bigr\}_j\bigr|^2 +\abs{α_j}^4 <\infty \] We also prove that \[ \int (1-\cosθ)^2 \log w(θ) \f{dθ}{2π} >-\infty \] if and only if \[ \sum_{j=0}^\infty \abs{α_{j+2}-2α_{j+1} +α_j}^2 + \abs{α_j}^6 <\infty \]

math-ph↗
arXiv2004

Sum Rules for Jacobi Matrices and Their Applications to Spectral Theory

We discuss the proof of and systematic application of Case's sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices J for which J-J_0 is Hilbert--Schmidt, and a proof of Nevai's conjecture that the Szego condition holds if J-J_0 is trace class.

math-ph↗
arXiv2004

On a theorem of Kac and Gilbert

We prove a general operator theoretic result that asserts that many multiplicity two selfadjoint operators have simple singular spectrum.

math.SP↗
arXiv2004

Limits of Zeros of Orthogonal Polynomials on the Circle

We prove that there is a universal measure on the unit circle such that any probability measure on the unit disk is the limit distribution of some subsequence of the corresponding orthogonal polynomials. This follows from an extension of a result of Alfaro and Vigil (which answered a question of Turán): namely, for $n<N$, one can freely prescribe the $n$-th polynomial and $N-n$ zeros of the $N$-th one. We shall also describe all possible limit sets of zeros within the unit disk.

math.SP↗
arXiv2004

The sharp form of the strong Szego theorem

Let $f$ be a function on the unit circle and $D_n(f)$ be the determinant of the $(n+1)\times (n+1)$ matrix with elements $\{c_{j-i}\}_{0\leq i,j\leq n}$ where $c_m =\hat f_m\equiv \int e^{-imθ} f(θ) \f{dθ}{2π}$. The sharp form of the strong Szegő theorem says that for any real-valued $L$ on the unit circle with $L,e^L$ in $L^1 (\f{dθ}{2π})$, we have \[ \lim_{n\to\infty} D_n(e^L) e^{-(n+1)\hat L_0} = \exp \biggl(\sum_{k=1}^\infty k\abs{\hat L_k}^2\biggr) \] where the right side may be finite or infinite. We focus on two issues here: a new proof when $e^{iθ}\to L(θ)$ is analytic and known simple arguments that go from the analytic case to the general case. We add background material to make this article self-contained.

math.SP↗
arXiv2003

Sturm Oscillation and Comparison Theorems

This is a celebratory and pedagogical discussion of Sturm oscillation theory. Included is the discussion of the difference equation case via determinants and a renormalized oscillation theorem of Gesztesy, Teschl, and the author.

math.SP↗
arXiv2003

Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schrödinger Operators

We announce three results in the theory of Jacobi matrices and Schrödinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schrödinger operator $-\f{d^2}{dx^2} +V(x)$ on $L^2 (0,\infty)$ with $V\in L^2 (0,\infty)$ and $u(0)=0$ boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szegő asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a given rate.

math.SP↗
arXiv2002

Variational Estimates for Discrete Schrödinger Operators with Potentials of Indefinite Sign

Let $H$ be a one-dimensional discrete Schrödinger operator. We prove that if $σ_{\ess} (H)\subset [-2,2]$, then $H-H_0$ is compact and $σ_{\ess}(H)=[-2,2]$. We also prove that if $H_0 + \frac14 V^2$ has at least one bound state, then the same is true for $H_0 +V$. Further, if $H_0 + \frac14 V^2$ has infinitely many bound states, then so does $H_0 +V$. Consequences include the fact that for decaying potential $V$ with $\liminf_{|n|\to\infty} |nV(n)| > 1$, $H_0 +V$ has infinitely many bound states; the signs of $V$ are irrelevant. Higher-dimensional analogues are also discussed.

math-ph↗
arXiv2002

Zeros of orthogonal polynomials on the real line

Let $p_n(x)$ be orthogonal polynomials associated to a measure $dμ$ of compact support in $R$. If $E\not\in supp(dμ)$, we show there is a $δ>0$ so that for all $n$, either $p_n$ or $p_{n+1}$ has no zeros in $(E-δ, E+δ)$. If $E$ is an isolated point of $supp(dμ)$, we show there is a $δ$ so that for all $n$, either $p_n$ or $p_{n+1}$ has at most one zero in $(E-δ, E+δ)$. We provide an example where the zeros of $p_n$ are dense in a gap of $supp(dμ)$.

math.CA↗
arXiv2002

Bound States and the Szego Condition for Jacobi Matrices and Schrodinger Operators

For Jacobi matrices with a_n = 1+(-1)^n alpha n^{-gamma}, b_n = (-1)^n beta n^{-gamma}, we study bound states and the SzegHo condition. We provide a new proof of Nevai's result that if gamma > 1/2, the Szego condition holds, which works also if one replaces (-1)^n by cos(mu n). We show that if alpha = 0, beta not equal to 0, and gamma < 1/2, the Szego condition fails. We also show that if gamma = 1, alpha and beta are small enough (beta^2 + 8 alpha^2 < 1/24 will do), then the Jacobi matrix has finitely many bound states (for alpha = 0, beta large, it has infinitely many).

math-ph↗
arXiv2002

Sum Rules and the Szego Condition for Orthogonal Polynomials on the Real Line

We study the Case sum rules, especially $C_0$, for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohat's theorem to cases with an infinite point spectrum and a proof that if $\lim n (a_n -1)=α$ and $\lim nb_n =β$ exist and $2α<\absβ$, then the Szegő condition fails.

math-ph↗
arXiv2001

Lieb-Thirring Inequalities for Jacobi Matrices

For a Jacobi matrix J on l^2(Z_+) with Ju(n)=a_{n-1} u(n-1) + b_n u(n) + a_n u(n+1), we prove that \sum_{|E|>2} (E^2 -4)^{1/2} \leq \sum_n |b_n| + 4\sum_n |a_n -1|. We also prove bounds on higher moments and some related results in higher dimension.

math-ph↗
arXiv2000

A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure

We continue the study of the A-amplitude associated to a half-line Schrodinger operator, -d^2/dx^2+ q in L^2 ((0,b)), b <= infinity. A is related to the Weyl-Titchmarsh m-function via m(-κ^2) =-κ- \int_0^a A(α) e^{-2ακ} dα+O(e^{-(2a -ε)κ}) for all ε> 0. We discuss five issues here. First, we extend the theory to general q in L^1 ((0,a)) for all a, including q's which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure ρ: A(α) = -2\int_{-\infty}^\infty λ^{-\frac12} \sin (2α\sqrtλ)\, dρ(λ) (since the integral is divergent, this formula has to be properly interpreted). Third, we provide a Laplace transform representation for m without error term in the case b<\infty. Fourth, we discuss m-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can compute A exactly.

math.SP↗
arXiv1999

A new approach to inverse spectral theory, I. Fundamental formalism

We present a new approach (distinct from Gel'fand-Levitan) to the theorem of Borg-Marchenko that the m-function (equivalently, spectral measure) for a finite interval or half-line Schrödinger operator determines the potential. Our approach is an analog of the continued fraction approach for the moment problem. We prove there is a representation for the m-function m(-κ^2) = -κ- \int_0^b A(α) e^{-2ακ}\, dα+ O(e^{-(2b-\varepsilon)κ}). A on [0,a] is a function of q on [0,a] and vice-versa. A key role is played by a differential equation that A obeys after allowing x-dependence: \frac{\partial A}{\partial x} = \frac{\partial A}{\partial α} + \int_0^αA(β, x) A(α-β, x)\, dβ. Among our new results are necessary and sufficient conditions on the m-functions for potentials q_1 and q_2 for q_1 to equal q_2 on [0,a].

math.SP↗