Gap-length mapping for periodic Jacobi matrices

We construct a real analytic isomorphism between periodic Jacobi operators and the spectral data formed by the gap lengths, the distances between the Dirichlet eigenvalues and the center of the corresponding gap, and some signs. This proves the uniqueness of the solution of the inverse problem and gives a characterization of the solution. Moreover, two-sided a priori estimates of periodic Jacobi operators in terms of gap lengths are obtained. Dedicated to the memory of B. M. Levitan     

Almost periodic Jacobi matrices associated with Julia sets for polynomials

Let be the Jacobi matrix associated with polynomialT(z) of degreeN2. The spectrum of is the Julia set associated withT(z) which in many cases is a Cantor set. Let ⁽¹⁾ denote the result of omitting the first row and column ofJ. Then it is shown that the spectrum of ⁽¹⁾ may be purely discrete.It is also shown that forT(z)= ^NC_N(z/) for > , whereC _N is a Chebychev polynomial the coefficients of and ⁽¹⁾ are limit periodic extending the work of Bellissard, Bessis, and Moussa (Phys. Rev. Lett.49, 701–704 (1982)).Supported in part by N.S.F. grant DMS-8401609Supported in part by N.S.F. grant MCS-8203325     

Inverse spectral theory for random Jacobi matrices

We give necessary and sufficient conditions for a Herglotz function to be thew-function of a random stationary Jacobi matrix.This paper is dedicated to the memory of Mark Kac.     

Kotani theory for one dimensional stochastic Jacobi matrices

We consider families of operators,H _ω, on ?₂ given by (H _ω u)(n)=u(n+1)+u(n?1)+V _ω(n)u(n), whereV _ω is a stationary bounded ergodic sequence. We prove analogs of Kotani's results, including that for a.e. ω,σ_ac(H _ω) is the essential closure of the set ofE where γ(E) the Lyaponov index, vanishes and the result that ifV _ω is non-deterministic, then σ_ac is empty.     

A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants

We study ergodic Jacobi matrices onl ²(Z), and prove a general theorem relating their a.c. spectrum to the spectra of periodic Jacobi matrices, that are obtained by cutting finite pieces from the ergodic potential and then repeating them. We apply this theorem to the almost Mathieu operator: (H _{, ,} u)(n)=u(n+1)+u(n–1)+ cos(2n+)u(n), and prove the existence of a.c. spectrum for sufficiently small , all irrational 's, and a.e. . Moreover, for 0<2 and (Lebesgue) a.e. pair , , we prove the explicit equality of measures: |_ac|=||=4 –2.Work partially supported by the US-Israel BSF     

Localization in general one dimensional random systems,I. Jacobi matrices

We consider random discrete Schrödinger operators in a strip with a potentialV (n, ) (n a label in and a finite label across the strip) andV an ergodic process. We prove thatH ₀+V has only point spectrum with probability one under two assumptions: (1) Theconditional distribution of {V (n,)} _n=0,1;all conditioned on {V } _n0,1;all has an absolutely continuous component with positive probability. (2) For a.e.E, no Lyaponov exponent is zero.Research partially supported by USNSF grant MCS-81-20833     

非线性波动方程的Jacobi椭圆函数包络周期解

应用Jacobi椭圆函数展开法求得了一类非线性波方程的包络周期解,而且用这种方法得到的周期解在一定条件下可以退化为包络冲击波解或包络孤立波解 关键词： Jacobi椭圆函数 非线性方程 包络周期解 包络孤立波解     

扩展的Jacobi椭圆函数展开法和Zakharov方程组的新的精确周期解

对Jacobi椭圆函数展开法进行了扩展，且利用这一方法求出了Zakharov方程组的一系列新的精确周期解，在极限情况下可得到相应的孤波解，补充了前面研究的结果. 关键词： Jacobi椭圆函数展开法 非线性发展方程 精确解 周期解     

Dynamical ordering of symmetric non-Birkhoff periodic points in reversible monotone twist mappings

We study the coexistence of symmetric non-Birkhoff periodic orbits of C(1) reversible monotone twist mappings on the cylinder. We prove the equivalence of the existence of non-Birkhoff periodic orbits and that of transverse homoclinic intersections of stable and unstable manifolds of the fixed point. We derive the positional relation of symmetric Birkhoff and non-Birkhoff periodic orbits and obtain the dynamical ordering of symmetric non-Birkhoff periodic orbits. An extension of the Sharkovskii ordering to two-dimensional mappings has been carried out. In the proof of various properties of the mappings, reversibility plays an essential role. (c) 2002 American Institute of Physics.     

Statistically periodic processes and toeplitz(α by β) matrices

If each element in a Toeplitz matrix is replaced by an by matrix and the original constraints preserved, the result is a doubly infinite matrix with periodic structure called a toeplitz_{( by)} matrix. Such matrices are a basic tool for describing, generating, estimating, filtering, synchronizing, and analyzing information-theoretic functions for statistically periodic processes.     

Inverse resonance scattering for Jacobi operators

The Jacobi operator (Jf) _n = a _n−1 f _n−1 +a _n f _n+1 + b _n f _n on ℤ with real finitely supported sequences (a _n − 1) _n∈ℤ and (b _n ) _n∈ℤ is considered. The inverse problem for two mappings (including their characterization): (a _n , b _n , n ∈ ℤ) → {the zeros of the reflection coefficient} and (a _n , b _n , n ∈ ℤ) → {the eigenvalues and the resonances} is solved. All Jacobi operators with the same eigenvalues and resonances are also described.     

Multi-Scale Jacobi Method for Anderson Localization

We prove that the fluctuations of mesoscopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green’s function for the associated Jacobi matrices.As a particular consequencewe obtain a central limit theorem for the modified Jacobi Unitary Ensembles on all mesoscopic scales.     

Spectral Estimates for Periodic Jacobi Matrices

 We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on ℓ²(ℤ) of the form (Hψ) _n =a _n−1 ψ _n−1 +b _nψ _n +a _nψ _n+1 , where a _n=a _n+q and b _n=b _n+q are periodic sequences of real numbers. The results are based on a study of the quasimomentum k(z) corresponding to H. We consider k(z) as a conformal mapping in the complex plane. We obtain the trace identities which connect integrals of the Lyapunov exponent over the gaps with the normalised traces of powers of H. Received: 17 April 2002 / Accepted: 1 October 2002 Published online: 13 January 2003 Communicated by B. Simon     

Integrable mappings for noncommutative objects

The method of integrable mappings is generalized to the noncommutative case. Hierarchies of integrable systems corresponding to the noncommutative Darboux-Toda substitution in the two-dimensional spaces and superspaces are constructed.     

Jacobi no-core shell model for p-shell hypernuclei

The European Physical Journal A - The ternary cluster decay of heavy nuclei has been observed in several experiments with binary coincidences between two fragments using detector telescopes (the...     

R-separation for the Hamilton—Jacobi equation

We present a non-trivial example of the occurrence of R-separation for the Hamilton—Jacobi equation on a complex Riemannian manifold. In our example the R-separation functions depends on a free parameter, this gives rise to a one-parameter family of R-separable solutions of the corresponding Helmholtz equation.     

On Jacobi fields

We define curves on a Riemannian manifold as integrals of generalized Jacobi fields. We show that the force term that deviates the trajectory from the geodesic motion can be constructed as a functional of the metric tensor. These curves can be interpreted as particles (observers) coupled nonminimally with gravitation that can provide a class of residual observers for the inevitable singularity—as shown in the text.This essay received an honorable mention (1976) from the Gravity Research Foundation-Ed.     

The Jacobi map

This paper defines nth order Jacobi fields to be solutions to a second-order nonlinear differential equation defined by the Jacobi map. nth order Jacobi fields arise naturally as acceleration vector fields of geodesic variations. As a main theorem we prove necessity and sufficiency conditions for an nth order Jacobi field to be the acceleration vector field of a variation of geodesics normal to a submanifold. An m geodesic, m ≥ 2, is a smooth curve whose mth covariant derivative vanishes. We prove an index theorem giving bounds for the total m focal multiplicity along an m geodesic m normal to a submanifold in a flat manifold.