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1.
S. Kupin 《Journal of Functional Analysis》2005,227(1):1-29
In this article, we relate the properties of elements of a Jacobi matrix from certain class to the properties of its spectral measure. The main tools we use are the so-called sum rules introduced by Case in [Orthogonal polynomials from the viewpoint of scattering theory, J. Math. Phys. 15 (1974) 2166-2174; Orthogonal polynomials, II. J. Math. Phys. 16 (1975) 1435-1440]. Later, the sum rules were efficiently applied by Killip-Simon [Sum rules for Jacobi matrices and their applications to spectral theory. Ann. Math. 158 (2003) 253-321] to the spectral analysis of Jacobi matrices. We use a modification of the method that permits us to work with sum rules of higher orders. As a corollary of the main theorem, we obtain a counterpart of a result of Molchanov-Novitskii-Vainberg [First KdV integrals and absolutely continuous spectrum for 1-D Schrödinger operator, Comm. Math. Phys. 216 (2001) 195-213] for a “continuous” Schrödinger operator on a half-line. 相似文献
2.
Barry Simon 《Journal of Functional Analysis》2004,214(2):396-409
We prove a general canonical factorization for meromorphic Herglotz functions on the unit disk whose notable elements are that there is no restriction (other than interlacing) on the zeros and poles for their Blaschke product to converge and there is no singular inner function. We use this result to provide a significant simplification in the proof of Killip-Simon (Ann. Math. 158 (2003) 253) of their result characterizing the spectral measures of Jacobi matrices, J, with J−J0 Hilbert-Schmidt. We prove a nonlocal version of Case and step-by-step sum rules. 相似文献
3.
Andrej Zlatoš 《Journal of Functional Analysis》2005,225(2):371-382
Let Ej be the eigenvalues outside [-2,2] of a Jacobi matrix with an-1∈?2 and bn→0, and μ′ the density of the a.c. part of the spectral measure for the vector δ1. We show that if bn∉?4, bn+1-bn∈?2, then
4.
Sergey A. Denisov 《Proceedings of the American Mathematical Society》2004,132(3):847-852
We prove Rakhmanov's theorem for Jacobi matrices without the additional assumption that the number of bound states is finite. This result solves one of Nevai's open problems.
5.
Recently Xu [13] proposed a new algorithm for computing a Jacobi matrix of order 2n with a given n×n leading principal submatrix and with 2n prescribed eigenvalues that satisfy certain conditions. We compare this algorithm to a scheme proposed by Boley and Golub
[2], and discuss a generalization that allows the conditions on the prescribed eigenvalues to be relaxed.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
6.
Peter Berkics 《Linear and Multilinear Algebra》2017,65(10):2114-2123
AbstractIn this note, I gather different definitions of the parallel sum of matrices, including the ones which work in infinite dimension as well. I describe the algebraic and analytic properties of this matrix operation, some interesting inequalities and its relation with other operator means. 相似文献
7.
V. V. Zudilin 《Mathematical Notes》1996,60(6):642-648
Algebraic properties of functional matrices arising in the constuction of graded Padé approximations are established. This
construction plays an important role in the theory of transcendental numbers.
Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 851–860, December, 1996.
This research was partially supported by the Russian Foundation for Basic Research under grant No. 94-01-00739. 相似文献
8.
Yinghong Xu 《Linear algebra and its applications》2012,436(9):3618-3633
In this paper, we investigate the properties of a special kind of periodic Jacobi matrices. We show that the solution of the inverse problem for periodic Jacobi matrices is unique if and only if the matrix is of that special kind. Moreover, we present an algorithm to construct the solution of that inverse problem and give some illustrative examples. Finally, we perform a stability analysis of the algorithm. 相似文献
9.
The spectrum σ of a non-negative Jacobi matrix J is characterized. If J is also required to be irreducible, further conditions on σ are needed, some of which are explored. 相似文献
10.
S. Kupin 《Proceedings of the American Mathematical Society》2004,132(5):1377-1383
Let be a Jacobi matrix with elements on the main diagonal and elements on the auxiliary ones. We suppose that is a compact perturbation of the free Jacobi matrix. In this case the essential spectrum of coincides with , and its discrete spectrum is a union of two sequences 2, x^-_j<-2$">, tending to . We denote sequences and by and , respectively.
The main result of the note is the following theorem.
Theorem. Let be a Jacobi matrix described above and be its spectral measure. Then if and only if
-\infty,\qquad {ii)} \sum_j(x^\pm_j\mp2)^{7/2}<\infty. \end{displaymath}">
11.
The Green's function method used by Case and Kac is extended to include unbounded Jacobi matrices. As a first application an upper bound on the number of eigenvalues is calculated, using the method of Bargmann. Another bound is found using the Birman-Schwinger argument, which is valid for matrix orthogonal polynomials. 相似文献
12.
L. N. Vaserstein 《Linear and Multilinear Algebra》1987,21(3):261-270
A theorem of Lagrange says that every natural number is the sum of 4 squares. M. Newman proved that every integral n by n matrix is the sum of 8 (-1)n squares when n is at least 2. He asked to generalize this to the rings of integers of algebraic number fields. We show that an n by n matrix over a a commutative R with 1 is the sum of squares if and only if its trace reduced modulo 2Ris a square in the ring R/2R. It this is the case (and n is at least 2), then the matrix is the sum of 6 squares (5 squares would do when n is even). Moreover, we obtain a similar result for an arbitrary ring R with 1. Answering another question of M. Newman, we show that every integral n by n matrix is the sum of ten k-th powers for all sufficiently large n. (depending on k). 相似文献
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16.
《Journal of Computational and Applied Mathematics》2001,127(1-2):17-65
Complex Jacobi matrices play an important role in the study of asymptotics and zero distribution of formal orthogonal polynomials (FOPs). The latter are essential tools in several fields of numerical analysis, for instance in the context of iterative methods for solving large systems of linear equations, or in the study of Padé approximation and Jacobi continued fractions. In this paper we present some known and some new results on FOPs in terms of spectral properties of the underlying (infinite) Jacobi matrix, with a special emphasis to unbounded recurrence coefficients. Here we recover several classical results for real Jacobi matrices. The inverse problem of characterizing properties of the Jacobi operator in terms of FOPs and other solutions of a given three-term recurrence is also investigated. This enables us to give results on the approximation of the resolvent by inverses of finite sections, with applications to the convergence of Padé approximants. 相似文献
17.
We apply the methods of classical approximation theory (extreme properties of polynomials) to study the essential support of the absolutely continuous spectrum of Jacobi matrices. First, we prove an upper bound on the measure of which takes into account the value distribution of the diagonal elements, and implies the bound due to Deift–Simon and Poltoratski–Remling.Second, we generalise the differential inequality of Deift–Simon for the integrated density of states associated with the absolutely continuous spectrum to general Jacobi matrices. 相似文献
18.
We study spectrum inclusion regions for complex Jacobi matrices that are compact perturbations of the discrete Laplacian. The condition sufficient for the lack of a discrete spectrum for such matrices is given.
19.
Ole H. Hald 《Linear algebra and its applications》1976,14(1):63-85
It is proved that a real symmetric tridiagonal matrix with positive codiagonal elements is uniquely determined by its eigenvalues and the eigenvalues of the largest leading principal submatrix, and can be constructed from these data. The matrix depends continuously on the data, but because of rounding errors the investigated algorithm might in practice break down for large matrices. 相似文献
20.
We use the classical results of Baxter and Golinskii–Ibragimov to prove a new spectral equivalence for Jacobi matrices on . In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences and find necessary and sufficient conditions on the spectral measure such that and lie in or for s1. 相似文献