Spectral properties of Jacobi matrices by asymptotic analysis (original) (raw)
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AMERICAN MATHEMATICAL SOCIETY, 1982
We consider the operator// on l2(Z) depending upon three parameters, X, a, 0, ... (1) [#(X, a, 0)u] (n) = u(n + 1) + u(n - 1) + X cos(2iran + 6)u(n). ... In this note we will sketch the proof of the following result whose detailed proof will appear elsewhere [3]. ... THEOREM 1. Fix a, an ...
Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices
Journal of Computational and Applied Mathematics, 2007
In this article we calculate the asymptotic behaviour of the point spectrum for some special self-adjoint unbounded Jacobi operators J acting in the Hilbert space l 2 = l 2 (N). For given sequences of positive numbers n and real q n the Jacobi operator is given by J = SW + W S * + Q, where Q = diag(q n) and W = diag(n) are diagonal operators, S is the shift operator and the operator J acts on the maximal domain. We consider a few types of the sequences {q n } and { n } and present three different approaches to the problem of the asymptotics of eigenvalues of various classes of J's. In the first approach to asymptotic behaviour of eigenvalues we use a method called successive diagonalization, the second approach is based on analytical models that can be found for some special J's and the third method is based on an abstract theorem of Rozenbljum.
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Opuscula Mathematica, 2014
The spectral properties and the asymptotic behaviour of the discrete spectrum for a special class of infinite tridiagonal matrices are given. We derive the asymptotic formulae for eigenvalues of unbounded complex Jacobi matrices acting in l 2 (N).
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We study the spectrum of periodic Jacobi matrices. We concentrate on the case of slowly oscillating diagonal terms and study the behaviour of the zeros of the associated orthogonal polynomials in the spectral gap. We find precise estimates for the distance from single eigenvalues of truncated matrices in the spectral gap to the diagonal entries of the matrix. We include a brief numerical example to show the exactness of our estimates.
Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices
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The aim of this paper is to find asymptotic formulas for eigenvalues of self-adjoint discrete operators in l 2 (N) given by some infinite symmetric Jacobi matrices. The approach used to calculate an asymptotic behaviour of eigenvalues is based on method of diagonalization, Janas and Naboko's lemma [J. Janas, S. Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal. 36 (2) (2004) 643-658] and the Rozenbljum theorem [G.V. Rozenbljum, Near-similarity of operators and the spectral asymptotic behaviour of pseudodifferential operators on the circle, (Russian) Trudy Maskov. Mat. Obshch. 36 (1978) 59-84]. The asymptotic formulas are given with use of eigenvalues and determinants of finite tridiagonal matrices.
Spectral analysis of unbounded Jacobi operators with oscillating entries
Studia Mathematica, 2012
We describe the spectra of Jacobi operators J with some irregular entries. We divide R into three "spectral regions" for J and using the subordinacy method and asymptotic methods based on some particular discrete versions of Levinson's theorem we prove the absolute continuity in the first region and the pure pointness in the second. In the third region no information is given by the above methods, and we call it the "uncertainty region". As an illustration, we introduce and analyse the O&P family of Jacobi operators with weight and diagonal sequences {wn}, {qn}, where wn = n α + rn, 0 < α < 1 and {rn}, {qn} are given by "essentially oscillating" weighted Stolz D 2 sequences, mixed with some periodic sequences. In particular, the limit point set of {rn} is typically infinite then. For this family we also get extra information that some subsets of the uncertainty region are contained in the essential spectrum, and that some subsets of the pure point region are contained in the discrete spectrum.
Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique
Studia Mathematica, 2018
For an arbitrary Hermitian period-T Jacobi operator, we assume a perturbation by a Wigner-von Neumann type potential to devise subordinate solutions to the formal spectral equation for a (possibly infinite) real set, S, of the spectral parameter. We employ discrete Levinson type techniques to achieve this, which allow the analysis of the asymptotic behaviour of the solution. This enables us to construct infinitely many spectral singularities on the absolutely continuous spectrum of the periodic Jacobi operator, which are stable with respect to an l 1-perturbation. An analogue of the quantisation conditions from the continuous case appears, relating the frequency of the oscillation of the potential to the quasi-momentum associated with the purely periodic operator.
2005
For all hyperbolic polynomials we proved in [11] a Lipschitz estimate of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron-Frobenius-Ruelle operators associated to hyperbolic polynomial dynamics (with real Julia set). Here we prove that for all sufficiently hyperbolic polynomials this estimate becomes exponentially better when the dimension of the Jacobi matrix grows. In fact, our main result asserts that a certain natural non-linear operator on Jacobi matrices built by a hyperbolic polynomial with real Julia set is a contraction in operator norm if the polynomial is sufficiently hyperbolic. This allows us to get for such polynomials the solution of a problem of Bellissard, in other words, to prove the limit periodicity of the limit Jacobi matrix. This fact does not require the iteration of the same fixed polynomial, and therefore it gives a wide class of limit periodic Jacobi matrices with singular continuous spectrum.
Inverse spectral theory for Jacobi matrices and their almost periodicity
1994
In this paper we consider the inverse problem for bounded Jacobi matrices with nonempty absolutely continuous spectrum and as an application show the almost periodicity of some random Jacobi matrices. We do the inversion in two different ways. In the general case we use a direct method of reconstructing the Green functions. In the special case where we show the almost periodicity, we use an alternative method using the trace formula for points in the orbit of the matrices under translations. This method of reconstruction involves analyzing the Abel-Jacobi map and solving of the Jacobi inversion problem associated with an infinite genus Riemann surface constructed from the spectrum. Contents 1 Introduction 2 1.1 Ideas, strategies and limitations : : : : : : : : : : : : : : : : : 4 2 Inverse Spectral Theory 8 3 Interpolation theorem 18 4 Analysis on a Riemann surface 32 4.1 The Riemann Surface : : : : : : : : : : : : : : : : : : : : : : : 32 4.2 The Abel-Jacobi map : : : : : : : : :...