String Jacobi Matrices with Very Fast Decreasing Weights

In this work we investigate a class of string, compact Jacobi matrices with very rapidly decreasing weights. In particular, we have found Hellinger-Weyls function representation as ratio of two entire functions with known Taylor series representation. The asymptotical estimations of zeroes of this entire functions and of eigenvalues of a corresponding operator are found. The application of this theory to Rogers-Ramanujans functions is considered.Submitted: November 18, 2002 Revised: October 30, 2003     

Absolute Continuity for Unbounded Jacobi Matrices with Constant Row Sums

We investigate the spectral properties of a class of Jacobi matrices in which the subdiagonal entries are quadratics and the row sums are constants.     

Spectral Density of Jacobi Matrices with Small Deviations

We present several new asymptotic trace formulas for Jacobi matrices whose coefficients satisfy a small deviation condition. Our results extend most of the existing trace formulas for Jacobi matrices.     

Spectral Analysis of Selfadjoint Jacobi Matrices with Periodically Modulated Entries

This paper deals with the spectral analysis of a class of selfadjoint unbounded Jacobi matrices J with modulated entries. The entries have the form of smooth sequences that increase to infinity multiplied by proper periodic sequences. For this class criteria for pure absolute continuity of the spectrum or its discreteness, and the asymptotics of generalized eigenvectors of J, are given. Some examples illustrating the stability zones of spectral structure are presented.     

Decay Bounds on Eigenfunctions and the Singular Spectrum of Unbounded Jacobi Matrices

Bounds on the exponential decay of generalized eigenfunctionsof bounded and unbounded selfadjoint Jacobi matrices in are established. Two cases are considered separatelyand lead to different results: (i) the case in which the spectralparameter lies in a general gap of the spectrum of the Jacobimatrix and (ii) the case of a lower semibounded Jacobi matrixwith values of the spectral parameter below the spectrum. Itis demonstrated by examples that both results are sharp. Weapply these results to obtain a "many barriers-type" criterionfor the existence of square-summable generalized eigenfunctionsof an unbounded Jacobi matrix at almost every value of the spectralparameter in suitable open sets. In particular, this leads toexamples of unbounded Jacobi matrices with a spectral mobilityedge, i.e., a transition from purely absolutely continuous spectrumto dense pure point spectrum.     

Unbounded Jacobi Matrices with a Few Gaps in the Essential Spectrum: Constructive Examples

We give explicit examples of unbounded Jacobi operators with a few gaps in their essential spectrum. More precisely a class of Jacobi matrices whose absolutely continuous spectrum fills any finite number of bounded intervals is considered. Their point spectrum accumulates to +?? and ???. The asymptotics of large eigenvalues is also found.     

Spectral Properties of Two-slanted Matrices

For two-slanted matrices, there is shown the close connection between their spectral properties and the zeros of their corresponding symbols. The results are applied to two-scale difference equations.     

Spectral Inclusion Properties of Unbounded Hamiltonian Operators

In this paper, the authors investigate the spectral inclusion properties of the quadratic numerical range for unbounded Hamiltonian operators. Moreover, some examples are presented to illustrate the main results.     

Spectral Measures and Jacobi Matrices Related to Laguerre-Type Systems of Orthogonal Polynomials

This paper considers systems of Laguerre-type orthogonal polynomials for which the corresponding Jacobi matrices represent unbounded self-adjoint operators which are bounded above or below. Under appropriate assumptions on the coefficient sequences in the recursion formula, results are obtained on the uniform boundedness of the polynomials on bounded intervals, the absence of eigenvalues for the corresponding operator, and the absolute continuity of the measure of orthogonality. Date received: September 7, 1995. Date revised: April 17, 1996.     

Some Spectral Properties of Infinite Band Matrices

For operators generated by a certain class of infinite band matrices we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying higher order recurrence relations. This enables us to describe some asymptotic behaviour of the corresponding systems of vector orthogonal polynomials. Finally, we provide some new convergence results for matrix Padé approximants.     

Some Spectral Properties of Metapositive Definite Matrice

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Localized Polynomial Frames on the Interval with Jacobi Weights

As is well known the kernel of the orthogonal projector onto the polynomials of degree n in L²(w_α,β, [−1, 1]), w_α,β(t) = (1 − t)^α(1 + t)^β, can be written in terms of Jacobi polynomials. It is shown that if the coefficients in this kernel are smoothed out by sampling a compactly supported C^∞ function then the resulting function has nearly exponential (faster than any polynomial) rate of decay away from the main diagonal. This result is used for the construction of tight polynomial frames for L²(w_α,β) with elements having almost exponential localization.     

概周期摆型方程无界解的存在性

对摆型方程x+Gx(x,t)=p(t),其中G(x,t)∈C1(R2)关于变量x是1周期的,并且sup(x,t)∈R2|Gx(x,t)|<+∞,limsupt→∞{supx∈R}=0,p(t)是平均值非零的概周期函数,证明了在柱面S1×R上方程具有无穷多的无界解.     

Unbounded Solutions of Almost Periodically Forced Pendulum-Type Equations

In this paper, we prove that the forced pendulum-type equation , where with 1-periodicity in x satisfies the conditions: and , possesses infinitely many unbounded solutions on a cylinder S ¹×R for any almost periodic function p(t) with nonvanishing mean value. Received October 7, 1997, Accepted June 9, 1999     

Complete Indefiniteness Tests for Jacobi Matrices with Matrix Entries

We study the problem concerning the maximality of the deficiency indices of operators generated by symmetric Jacobi matrices with matrix entries in the space . Effective conditions for the maximality of the deficiency indices are given in terms of entries of the Jacobi matrix. These conditions are new even in the scalar (one-dimensional) case.     

Transformation Operator for Jacobi Matrices with Asymptotically Periodic Coefficients

We construct the transformation operator for the scattering problem with a periodic background under the assumption that the coefficients of the perturbation have a first finite moment. By means of the Marchenko approach [Marchenko, V. (1986) Sturm–Liouville Operators and Applications. Birkhäuser, Basel, Switzerland] we derive an estimate on the kernel of this transformation operator that allow us to study the inverse problem solution in the prescribed class of perturbations.