David Damanik, Barry Simon
We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Jost solution. We also prove $L^2$ convergence of Szegő asymptotics on the spectrum.
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We consider Jacobi matrices whose essential spectrum is a finite union of closed intervals. We focus on Szego's theorem, Jost solutions, and Szego asymptotics for this situation. This announcement describes talks the authors gave at OPSFA 2007.
We show that the parameters $a_n, b_n$ of a Jacobi matrix have a complete asymptotic series $ a_n^2 -1 &= \sum_{k=1}^{K(R)} p_k(n) μ_k^{-2n} + O(R^{-2n}) b_n &= \sum_{k=1}^{K(R)} p_k(n) μ_k^{-2n+1} + O(R^{-2n}) $ where $1 < |μ_j| < R$ for $j\leq K(R)$ and all $R$ if and only if the Jost function, $u$, written in terms of $z$ (where $E=z+z^{-1}$) is an entire meromorphic function. We relate the poles of $u$ to the $μ_j$'s.
We present necessary and sufficient conditions on the Jost function for the corresponding Jacobi parameters $a_n -1$ and $b_n$ to have a given degree of exponential decay.
We introduce the Szegö class, Sz(E), for an arbitrary Parreau-Widom set E in R and study the dynamics of its elements under the left shift. When the direct Cauchy theorem holds on C\E, we show that to each J in Sz(E) there is a unique element J' in the isospectral torus, T_E, so that the left-shifts of J are asymptotic to the orbit {J'_m} on T_E. Moreover, we show that the ratio of the associated orthogonal polynomials has a limit, expressible in terms of Jost functions, as the degree n tends to infinity. This enables us to describe the large n behaviour of the orthogonal polynomials for every…
We give a simple and explicit description of the Bernstein-Szego type measures associated with Jacobi matrices which differ from the Jacobi matrix of the Chebyshev measure in finitely many entries. We also introduce a class of measures M which parametrizes the Jacobi matrices with exponential decay and for each element in M we define a scattering function. Using Banach algebras associated with increasing Beurling weights, we prove that the exponential decay of the coefficients in a Jacobi matrix is completely determined by the decay of the negative Fourier coefficients of the scattering functi…
We obtain matrix-valued Jost asymptotics for block Jacobi matrices under an L1-type condition on Jacobi parameters, and give a necessary and sufficient condition for an analytic matrix-valued function to be the Jost function of a block Jacobi matrix with exponentially converging parameters. This establishes the matrix-valued analogue of Damanik-Simon-II paper [6]. The above results allow us to fully characterize the matrix-valued Weyl-Titchmarsh m-functions of block Jacobi matrices with exponentially converging parameters.
We study semi-infinite Jacobi matrices $H=H_{0}+V$ corresponding to trace class perturbations $V$ of the "free" discrete Schrödinger operator $H_{0}$. Our goal is to construct various spectral quantities of the operator $H$, such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair $H_{0}$, $H$, the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials $P_{n}(z)$ associated to the Jacobi matrix $H $ as $n\to\infty$. In particular, we consider the case of $z$ inside the spe…
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…
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 \[ Q(x)=β^{-1} C^{\frac{1}β} \frac{Γ(\frac32)Γ(\frac{1}β}-\frac12)(2-x)^{\frac12 -\frac{1}β}}{Γ(\frac{1}β+1)}(1+O((2-x))) \]
We use a classical result of Gollinski and Ibragimov to prove an analog of the strong Szego theorem for Jacobi matrices on $l^2(\N)$. In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences and find necessary and sufficient conditions on the spectral measure such that $\sum_{k=n}^\infty b_k$ and $\sum_{k=n}^\infty (a_k^2 - 1)$ lie in $l^2_1$, the linearly-weighted $l^2$ space.
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).
Fredholm determinants, continued fractions, Jost and Evans functions for a Jacobi matrix associated with the 2D-Euler equations
本文针对二维环面欧拉方程单向流稳定性导出的二阶差分方程，证明了Fredholm行列式、Jost函数、Evans函数及前后向连分数五者相等且零点对应离散特征值，并据此改进了广义Kolmogorov流的不稳定性结果。
We consider the periodic Jacobi operator $J$ with finitely supported perturbations on the half-lattice. We describe all eigenvalues and resonances of $J$ and give their properties. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the Jost functions is one-to-one and onto, we show how the Jost functions can be reconstructed from the eigenvalues, resonances and the set of zeros of $S(ł)-1,$ where $S(ł)$ is the scattering matrix.
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_…
Wave operators for Jacobi matrices
研究满足泽格条件的雅可比矩阵的波算子，证明在温和附加条件下波算子的存在性和完备性。
We continue studying polynomials generated by the Szegő recursion when a finite number of Verblunsky coefficients lie outside the closed unit disk. We prove some asymptotic results for the corresponding orthogonal polynomials and then translate them to the real line to obtain the Szegő asymptotics for the resulting polynomials. The latter polynomials give rise to a non-symmetric tridiagonal matrix but it is a finite-rank perturbation of a symmetric Jacobi matrix.
This work is in a stream initiated by a paper of Killip and Simon [Ann. of Math. (2003)]. Using methods of Functional Analysis and the classical Szegö Theorem we prove sum rule identities in a very general form. Then, we apply the result to obtain new asymptotics for orthonormal polynomials.
Prüfer variables are a standard tool in spectral theory, developed originally for perturbations of the free Schrödinger operator. They were generalized by Kiselev, Remling, and Simon to perturbations of an arbitrary Schrödinger operator. We adapt these generalized Prufer variables to the setting of Jacobi and Szegő recursions. We present an application to random $L^2$ perturbations of Jacobi and CMV matrices, and an application to decaying oscillatory perturbations of periodic Jacobi and CMV matrices.
Let $\frak{e}\subset\mathbb{R}$ be a finite union of disjoint closed intervals. In the study of OPRL with measures whose essential support is $\frak{e}$, a fundamental role is played by the isospectral torus. In this paper, we use a covering map formalism to define and study this isospectral torus. Our goal is to make a coherent presentation of properties and bounds for this special class as a tool for ourselves and others to study perturbations. One important result is the expression of Jost functions for the torus in terms of theta functions.
We find and discuss asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with recurrence coefficients $a_{n}, b_{n}$. Our main goal is to consider the case where off-diagonal elements $a_{n}\to\infty$ as $n\to\infty$. Formulas obtained are essentially different for relatively small and large diagonal elements $b_{n}$. Our analysis is intimately linked with spectral theory of Jacobi operators $J$ with coefficients $a_{n}, b_{n}$ and a study of the corresponding second order difference equations. We introduce the Jost solutions $f_{n}(z)$, $n\geq -1$, of such equations by a condition for $…
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.
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 give…
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