In this paper, we prove a Szegö type limit theorem on $\ell^2(\ZZ^d)$. We consider operators of the form $H=Δ+V$, $V$ multiplication by a positive sequence $\{V(n), n \in \ZZ^d\}$ with $V(n) \rightarrow \infty, |n| \rightarrow \infty $
on $\ell^2(\ZZ^d)$ and $π_λ$ the orthogonal projection of $\ell^2(\mathbb{Z}^d)$ on to the space of eigenfunctions of $H$ with eigenvalues $\leq λ$. We take $B$ to be a pseudo difference operator of order zero with symbol $b(x,n), (x,n) \in \TT^d\times \ZZ^d$ and show that for nice functions $f$ $$ \lim_{λ\rightarrow \infty} Tr(f(π_λBπ_λ))/Tr(π_λ) = \lim_{λ\rightarrow \infty} \frac{1}{(2π)^d} \frac{\sum_{V(n) \leq λ} \int_{\TT^d} f(b(x,n)) ~ dx}{\sum_{V(n)\leqλ} 1}. $$
We disprove a conjecture of Simon for higher-order Szego theorems for orthogonal polynomials on the unit circle and propose a modified version of the conjecture.
The paper is a continuation of work [15] in which the general setting for analogs of the Szegö theorem for ergodic operators was given and several interesting cases were considered. Here we extend the results of [15] to a wider class of test functions and symbols which determine the Szegö-type asymptotic formula for the one-dimensional Schrodinger operator with random potential. We show that in this case the subleading term of the formula is given by a Central Limit Theorem in the spectral context, hence the term is asymptotically proportional to $L^{1/2}$, where $L$ is the length of the inter…
Given a representation of the circle group by semiclassical Fourier integral operators, we construct an algebra of semiclassical pseudodifferential operators that are a quantum analogue of the notion of symplectic cutting of Lerman, and we prove an associated Szegö limit theorem.
A Szegö-type theorem for Toeplitz operators was proved by Boutet de Monvel and Guillemin for general Toeplitz structures. We give a local version of this result in the setting of positive line bundles on compact symplectic manifolds.
Faddeev-Marchenko scattering for CMV matrices and the Strong Szego Theorem
B. Simon证明了具有Szego类Verblunsky系数的CMV矩阵的波算子的存在性，从而证明了散布函数的存在性。通常，当我们从散布函数开始时，没有希望恢复CMV矩阵，特别是因为它不包含任何关于（可能的）奇异测度的信息。我们感兴趣的主要点是逆散射问题的解决方案（心脏的Faddeev--Marchenko理论），也就是说，给充要条件上的某一类CMV矩阵，使限制这种对应关系（从矩阵的散布函数）是一对一。本文证明了逆散射的主要问题可以借助于三个重要的经典结果：Adamyan-Arov-Krein（AAK）理论、Helson-Szego定理和强Szego极限定理来解决。这些定理中的每一个都陈述了某些条件的等价性。实际上，每个定理，我们增加了一个有关的CMV逆散射问题的等价条件。
We consider canonical systems and investigate the Szegő class, which is defined via the finiteness of the associated entropy functional. Noting that the canonical system may be studied in a variety of gauges, we choose to work in the Arov gauge, in which we prove that the entropy integral is equal to an integral involving the coefficients of the canonical system. This sum rule provides a spectral theory gem in the sense proposed by Barry Simon.
The Szego-Widom theorem provides an expression for the determinant of block Toeplitz matrices in the asymptotic limit of large matrix dimension n. We show that the presence of zero modes, i.e, eigenvalues that vanish as α^n, |α|<1, when n \to \infty, require a modification of the Szego-Widom theorem. A new asymptotic expression for the determinant of a certain class of block Toeplitz matrices with one pair of zero modes is derived. The result is inspired by 1-dimensional topological superconductors, and the relation with the latter systems is discussed.
We study the weighted averages of resonance clusters for the hydrogen atom with a Stark electric field in the weak field limit. We prove a semiclassical Szego-type theorem for resonance clusters showing that the limiting distribution of the resonance shifts concentrates on the classical energy surface corresponding to a rescaled eigenvalue of the hydrogen atom Hamiltonian. This result extends Szego-type results on eigenvalue clusters to resonance clusters. There are two new features in this work: first, the study of resonance clusters requires the use of non self-adjoint operators, and second,…
We develop a new technique for computing a class of four-point correlation functions of heavy half-BPS operators in planar N=4 SYM theory which admit factorization into a product of two octagon form factors with an arbitrary bridge length. We show that the octagon can be expressed as the Fredholm determinant of the integrable Bessel operator and demonstrate that this representation is very efficient in finding the octagons both at weak and strong coupling. At weak coupling, in the limit when the four half-BPS operators become null separated in a sequential manner, the octagon obeys the Toda la…
We prove Szegő-type trace asymptotics for translation-invariant operators on polygons. More precisely, consider a Fourier multiplier $A=\mathcal{F}^\ast σ\mathcal{F}$ on $\mathsf{L}^2(\mathbb{R}^2)$ with a sufficiently decaying, smooth symbol $σ:\mathbb{C}\to\mathbb{C}$. Let $P\subset \mathbb{R}^2$ be the interior of a polygon and, for $L\geq 1$, define its scaled version $P_L:=L\cdot P$. Then we study the spectral asymptotics for the operator $A_{P_L}=χ_{P_L}Aχ_{P_L}$, the spatial restriction of $A$ onto $P_L$: for entire functions $h$ with $h(0)=0$ we provide a complete asymptotic expansion …
We prove a limiting eigenvalue distribution theorem (LEDT) for suitably scaled eigenvalue clusters around the discrete negative eigenvalues of the hydrogen atom Hamiltonian formed by the perturbation by a weak constant magnetic field. We study the hydrogen atom Zeeman Hamiltonian $H_V(h,B) = (1/2)( - i h {\mathbf \nabla} - {\mathbf A}(h))^2 - |x|^{-1}$, defined on $L^2 (R^3)$, in a constant magnetic field ${\mathbf B}(h) = {\mathbf \nabla} \times {\mathbf A}(h)=(0,0,ε(h)B)$ in the weak field limit $ε(h) \rightarrow 0$ as $h\rightarrow{0}$. We consider the Planck's parameter $h$ taking values a…
Gessel-Type Expansion for the Circular $β$-Ensemble and Central Limit Theorem for the Sine-$β$ Process for $β\le 2$
Institute for System Programming of the Russian Academy of Sciences（俄罗斯科学院系统编程研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）
