Andrej Zlatos
A Jacobi matrix with $a_n\to 1$, $b_n\to 0$ and spectral measure $ν'(x)dx + dν_{sing}(x)$ satisfies the Szeg\H o condition if $\int_{0}^π\ln \bigl[ ν'(2\cosθ) \bigr] dθ$ is finite. We prove that if $a_n = 1 + \frac α{n} + O(n^{-1-\eps})$ and $b_n = \frac β{n} + O(n^{-1-\eps})$ with $2α\ge |β|$ and $\eps>0$, then the corresponding matrix is Szeg\H o.
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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).
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 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.
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_…
Extending earlier work of Killip-Simon and Simon-Zlatos, we obtain sum rules for Jacobi matrices in which the a.c. part of the spectral measure and the eigenvalues of the matrix appear on opposite sides of the equation. We use these to obtain various results on divergence of certain Lieb-Thirring sums of eigenvalues and Szego-type integrals of the a.c. part of the spectral measure.
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。
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.
We present an informal review of results on asymptotics of orthogonal polynomials, stressing their spectral aspects and similarity in two cases considered. They are polynomials orthonormal on a finite union of disjoint intervals with respect to the Szego weight and polynomials orthonormal on R with respect to varying weights and having the same union of intervals as the set of oscillations of asymptotics. In both cases we construct double infinite Jacobi matrices with generically quasiperiodic coefficients and show that each of them is an isospectral deformation of another. Related results on …
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.
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.
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.
Szegő type correlations for two-dimensional outpost ensembles
本文研究了二维 outpost 集合的 Szegő 类相关性，通过再生核扩展了边缘相关性的研究，并探讨了外部点电荷的影响。
The two-body Coulomb Hamiltonian, when calculated in Coulomb-Sturmian basis, has an infinite symmetric tridiagonal form, also known as Jacobi matrix form. This Jacobi matrix structure involves a continued fraction representation for the inverse of the Green's matrix. The continued fraction can be transformed to a ratio of two $_{2}F_{1}$ hypergeometric functions. From this result we find an exact analytic formula for the matrix elements of the Green's operator of the Coulomb Hamiltonian.
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…
The two-dimensional Coulomb gas: fluctuations through a spectral gap
该研究针对逆温度β=2的带间隙径向对称二维库仑气体，推导了间隙附近相关物理量的大n精细渐近，揭示了粒子分布的振荡行为并给出显式表达。
We propose two ways for determining the Green's matrix for problems admitting Hamiltonians that have infinite symmetric tridiagonal (i.e. Jacobi) matrix form on some basis representation. In addition to the recurrence relation comming from the Jacobi-matrix, the first approach also requires the matrix elements of the Green's operator between the first elements of the basis. In the second approach the recurrence relation is solved directly by continued fractions and the solution is continued analytically to the whole complex plane. Both approaches are illustrated with the non-trivial but calcul…
In this paper we solve an arbitrary matrix Riemann-Hilbert (inverse monodromy) problem with quasi-permutation monodromy representations outside of a divisor in the space of monodromy data. This divisor is characterized in terms of the theta-divisor on the Jacobi manifold of an auxiliary compact Riemann surface realized as an appropriate branched covering of $\CP1$ . The solution is given in terms of a generalization of Szegö kernel on the Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. The isomonodromy tau-function of these solution…
Several examples of Jacobi matrices with an explicitly solvable spectral problem are worked out in detail. In all discussed cases the spectrum is discrete and coincides with the set of zeros of a special function. Moreover, the components of corresponding eigenvectors are expressible in terms of special functions as well. Our approach is based on a recently developed formalism providing us with explicit expressions for the characteristic function and eigenvectors of Jacobi matrices. This is done under an assumption of a simple convergence condition on matrix entries. Among the treated special …
We study the Hankel determinant generated by a singularly perturbed Jacobi weight $$ w(x,t):=(1-x^2)^α\mathrm{e}^{-\frac{t}{x^{2}}},\;\;\;\;\;\;x\in[-1,1],\;\;α>0,\;\;t\geq 0. $$ If $t=0$, it is reduced to the classical symmetric Jacobi weight. For $t>0$, the factor $\mathrm{e}^{-\frac{t}{x^{2}}}$ induces an infinitely strong zero at the origin. This Hankel determinant is related to the Wigner time-delay distribution in chaotic cavities. In the finite $n$ dimensional case, we obtain two auxiliary quantities $R_n(t)$ and $r_n(t)$ by using the ladder operator approach. We show that the Hankel de…
We describe an ensemble of (sparse) random matrices whose eigenvalues follow the Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature beta. Our approach combines elements from the theory of orthogonal polynomials on the unit circle with ideas from recent work of Dumitriu and Edelman. In particular, we resolve a question left open by them: find a tri-diagonal model for the Jacobi ensemble.
Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut $β$-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral density, and is therefore universal for a large class of models. This allows us to derive a closed-form expression for the limiting covariances of an arbitrary one-cut $β$-ensemble. As particular cases of the main result we consider the classical $β$-Gaussian, $β$-Wishart and $β$-Jacobi ensembles, for which we derive previously available results as well as new …
An invariant ensemble of $N\times N$ random matrices can be characterised by a joint distribution for eigenvalues $P(λ_1,\cdots,λ_N)$. The study of the distribution of linear statistics, i.e. of quantities of the form $L=(1/N)\sum_if(λ_i)$ where $f(x)$ is a given function, appears in many physical problems. In the $N\to\infty$ limit, $L$ scales as $L\sim N^η$, where the scaling exponent $η$ depends on the ensemble and the function $f$. Its distribution can be written under the form $P_N(s=N^{-η}\,L)\simeq A_{β,N}(s)\,\exp\big\{-(βN^2/2)\,Φ(s)\big\}$, where $β\in\{1,\,2,\,4\}$ is the Dyson inde…
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