Multithreshold Spectral Phase Transitions for a Class of Jacobi Matrices

Janas, Jan; Naboko, Serguei

doi:10.1007/978-3-0348-8323-8_13

Cited by 47 publications

(62 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Some kinds of this result were obtained by different methods in works [4,5,9,12]. Here we give independent, new and simple proof of it.…”

mentioning

confidence: 64%

On one condition of absolutely continuous spectrum for self-adjoint operators and its applications

Ianovich¹

2018

OpMath

View full text Add to dashboard Cite

Abstract. In this work the method of analyzing of the absolutely continuous spectrum for self-adjoint operators is considered. For the analysis it is used an approximation of a self-adjoint operator A by a sequence of operators An with absolutely continuous spectrum on a given interval [a, b] which converges to A in a strong sense on a dense set. The notion of equi-absolute continuity is also used. It was found a sufficient condition of absolute continuity of the operator A spectrum on the finite interval [a, b] and the condition for that the corresponding spectral density belongs to the class Lp [a, b] (p ≥ 1). The application of this method to Jacobi matrices is considered. As one of the results we obtain the following assertion: Under some mild assumptions, suppose that there exist a constant C > 0 and a positive function g(x) ∈ Lp [a, b] (p ≥ 1) such that for all n sufficiently large and almost all x ∈ [a, b] the estimate≤ bn(P 2 n+1 (x) + P 2 n (x)) ≤ C holds, where Pn(x) are 1st type polynomials associated with Jacobi matrix (in the sense of Akhiezer) and bn is a second diagonal sequence of Jacobi matrix. Then the spectrum of Jacobi matrix operator is purely absolutely continuous on [a, b] and for the corresponding spectral density f (x) we have f (x) ∈ Lp [a, b].

show abstract

“…Some kinds of this result were obtained by different methods in works [4,5,9,12]. Here we give independent, new and simple proof of it.…”

mentioning

confidence: 64%

On one condition of absolutely continuous spectrum for self-adjoint operators and its applications

Ianovich¹

2018

OpMath

View full text Add to dashboard Cite

show abstract

“…(iii) The assertion (b2) can be deduced from [6,7], see also [13]. (iv) The point (c1) follows from Theorem 8 of [5].…”

Section: Remark 22mentioning

confidence: 97%

“…(i) The point (a) is a particular case of Theorem 4.1 of [7] ensuring that J (a n , b n ) is essentially self-adjoint on l 2 0 and has no essential spectrum if lim n→∞ a n = +∞ and lim inf…”

Section: Remark 22mentioning

confidence: 99%

On the spectrum of periodic perturbations of certain unbounded Jacobi operators

Sahbani¹

2016

OpMath

View full text Add to dashboard Cite

show abstract

“…Subordination theory, formulated first by Gilbert and Pearson in [7] for one dimensional Shrödinger operators, where transferred to the discrete case by [22,34], became a very popular method used for spectral analysis of Jacobi operators in 2 (N) (see, e.g., [3][4][5][8][9][10][11][12][13][14][15][16][17][18][19][20][21][23][24][25][26][27]29,30,32,33]). One of the typical ways of employing the subordination theory for spectral studies is related to asymptotic studies, and it contains the following three basic steps:…”

Section: Introductionmentioning

confidence: 99%

“…Finding asymptotics of some solutions to the generalised eigenequation (generalised eigenvectors) for the Jacobi operator J. This is often achieved with the use of some discrete Levinson type theorems-see [3,5,9,[11][12][13]16,18,19,21,24,25,27,30,32]; 1 2. Reading some information on the existence of subordinate solution (see Definition (2.1)) from the asymptotic information on solutions; 3.…”

Section: Introductionmentioning

confidence: 99%

Non-Existence of Subordinate Solutions for Jacobi Operators in Some Critical Cases

Moszyński

2013

Integr. Equ. Oper. Theory

View full text Add to dashboard Cite

Abstract. We show a method of repairing some gaps in proofs of the absolute continuity of the spectrum of Jacobi operators. Such gaps have been found in several recent papers, dealing mainly with the so-called critical case (i.e., Jordan box case). We solve the problem by proving that the subordinate solution does not exist for many cases with two linearly independent generalised eigenvectors possessing "similar" asymptotic behaviour.Mathematics Subject Classification (2010). 47B25, 47B36, 47B39, 47A10, 39A11, 40G05.

show abstract

Multithreshold Spectral Phase Transitions for a Class of Jacobi Matrices

Cited by 47 publications

References 22 publications

On one condition of absolutely continuous spectrum for self-adjoint operators and its applications

On one condition of absolutely continuous spectrum for self-adjoint operators and its applications

On the spectrum of periodic perturbations of certain unbounded Jacobi operators

Non-Existence of Subordinate Solutions for Jacobi Operators in Some Critical Cases

Contact Info

Product

Resources

About