Limit points of eigenvalues of truncated tridiagonal operators

Ifantis, E. K.; Παναγόπουλος, Παναγιώτης

doi:10.1016/s0377-0427(00)00663-4

Cited by 12 publications

(10 citation statements)

References 11 publications

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“…. , a n−1 , u n ) for a sequence u in T. A recommendable choice is the sequence u = u w given in (19) that fixes a common eigenvalue w for all the finite matrices, because it permits us to control the only possible strong limit point that, according to Theorem 4.17, can lie outside supp . In this case Theorem 4.18 proves that the double limit points coincide with supp ∪ {w}, so the computation of the eigenvalues for pairs of consecutive matrices can be used to eliminate those weak limit points that are spurious points of supp .…”

Section: Proposition 415mentioning

confidence: 99%

Measures on the unit circle and unitary truncations of unitary operators

Cantero

Ituarte

Velázquez

2006

Journal of Approximation Theory

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In this paper, we obtain new results about the orthogonality measure of orthogonal polynomials on the unit circle, through the study of unitary truncations of the corresponding unitary multiplication operator, and the use of the five-diagonal representation of this operator.Unitary truncations on subspaces with finite co-dimension give information about the derived set of the support of the measure under very general assumptions for the related Schur parameters (a n ). Among other cases, we study the derived set of the support of the measure when lim n |a n+1 /a n | = 1, obtaining a natural generalization of the known result for the López class lim n a n+1 /a n ∈ T, lim n |a n | ∈ (0, 1).On the other hand, unitary truncations on subspaces with finite dimension provide sequences of unitary five-diagonal matrices whose spectra asymptotically approach the support of the measure. This answers a conjecture of L. Golinskii concerning the relation between the support of the measure and the strong limit points of the zeros of the para-orthogonal polynomials.Finally, we use the previous results to discuss the domain of convergence of rational approximants of Carathéodory functions, including the convergence on the unit circle.

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Section: Proposition 415mentioning

confidence: 99%

Measures on the unit circle and unitary truncations of unitary operators

Cantero

Ituarte

Velázquez

2006

Journal of Approximation Theory

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“…Information about Λ(T ) is of great importance not only from the point of view of operator theory, but also for the theory of continued fractions, orthogonal polynomials and numerical analysis (see [13] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

“…In the bounded case several sufficient conditions for Λ(T ) = σ(T ) can be found in the literature, but not much is known in the unbounded case (see [12,13], also [4,5] for related problems concerning non-symmetric tridiagonal operators, and [6] for extensions to unitary CMV operators). In particular, the authors of [12] established sufficient conditions for Λ(T ) = σ(T ) when b n is divergent, which regard the limits of some functions of a n , b n .…”

Section: Introductionmentioning

confidence: 99%

Self-adjointness of unbounded tridiagonal operators and spectra of their finite truncations

Petropoulou

Velázquez

2014

Journal of Mathematical Analysis and Applications

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This paper addresses two different but related questions regarding an unbounded symmetric tridiagonal operator: its self-adjointness and the approximation of its spectrum by the eigenvalues of its finite truncations. The sufficient conditions given in both cases improve and generalize previously known results. It turns out that, not only self-adjointness helps to study limit points of eigenvalues of truncated operators, but the analysis of such limit points is a key help to prove self-adjointness. Several examples show the advantages of these new results compared with previous ones. Besides, an application to the theory of continued fractions is pointed out.Keywords: unbounded symmetric operators, Jacobi matrices, self-adjointness, spectrum of an operator, limit points of eigenvalues, zeros of orthogonal polynomials, Jacobi continued fractions. 2000 MSC: 47B25, 47B36

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“…Second, at any time t ≥ 0, L(t) is a compact perturbation of the discrete free Schrrödinger operator L 0 that has a n ≡ 1 2 , b n ≡ 0, and σ L 0 = σ ac L 0 = [−1, 1]. As proven in [26], these two facts imply: Proposition 3.4 (Theorem 2.3, [26]). Let Z(L) denote the set of all limit points of ∞ N =1 σ L N .…”

Section: The Numerical Schemementioning

confidence: 83%

On the evolution of scattering data under perturbations of the Toda lattice

Bilman

Nenciu

2016

Physica D: Nonlinear Phenomena

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We present the results of an analytical and numerical study of the long-time behavior for certain Fermi-Pasta-Ulam (FPU) lattices viewed as perturbations of the completely integrable Toda lattice. Our main tools are the direct and inverse scattering transforms for doubly-infinite Jacobi matrices, which are well-known to linearize the Toda flow. We focus in particular on the evolution of the associated scattering data under the perturbed vs. the unperturbed equations. We find that the eigenvalues present initially in the scattering data converge to new, slightly perturbed eigenvalues under the perturbed dynamics of the lattice equation. To these eigenvalues correspond solitary waves that emerge from the solitons in the initial data. We also find that new eigenvalues emerge from the continuous spectrum as the lattice system is let to evolve under the perturbed dynamics.

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Limit points of eigenvalues of truncated tridiagonal operators

Cited by 12 publications

References 11 publications

Measures on the unit circle and unitary truncations of unitary operators

Measures on the unit circle and unitary truncations of unitary operators

Self-adjointness of unbounded tridiagonal operators and spectra of their finite truncations

On the evolution of scattering data under perturbations of the Toda lattice

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