Weyl–Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle

Gesztesy, Fritz; Zinchenko, Maxim

doi:10.1016/j.jat.2005.08.002

Cited by 57 publications

(112 citation statements)

References 34 publications

(50 reference statements)

Supporting

Mentioning

111

Contrasting

Order By: Relevance

“…The one difference is that since δ(x) is not in L 2 , we do not have the analog of (2.23). However, 18) which implies that 19) and that suffices to define an inverse transform on L 2 (…”

Section: Theorem 32 R(λ) Is Given By (314)mentioning

confidence: 99%

See 2 more Smart Citations

Equality of the Spectral and Dynamical Definitions of Reflection

Breuer

Ryckman

Simon

2009

Commun. Math. Phys.

View full text Add to dashboard Cite

For full-line Jacobi matrices, Schrödinger operators, and CMV matrices, we show that being reflectionless, in the sense of the well-known property of m-functions, is equivalent to a lack of reflection in the dynamics in the sense that any state that goes entirely to x = −∞ as t → −∞ goes entirely to x = ∞ as t → ∞. This allows us to settle a conjecture of Deift and Simon from 1983 regarding ergodic Jacobi matrices.

show abstract

“…The one difference is that since δ(x) is not in L 2 , we do not have the analog of (2.23). However, 18) which implies that 19) and that suffices to define an inverse transform on L 2 (…”

Section: Theorem 32 R(λ) Is Given By (314)mentioning

confidence: 99%

“…Two-sided CMV matrices were defined first in [40], although related objects appeared earlier in [3,10]. For further study, we mention [4,17,19,39].…”

Section: The CMV Casementioning

confidence: 99%

See 1 more Smart Citation

Equality of the Spectral and Dynamical Definitions of Reflection

Breuer

Ryckman

Simon

2009

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…10. We require a way to relate the extended CMV operator E with the two one-sided CMV operators that comprise its two halves.…”

Section: Main Tools and Backgroundmentioning

confidence: 99%

“…Now let us demonstrate that δ 2n + 3 is in S 2n, 2n + 1 . We consider (10) and (11) once more, and this time by eliminating the δ 2n , δ 2n + 1 terms, we get…”

Section: Proof Of Lemmamentioning

confidence: 99%

The Hölder continuity of spectral measures of an extended CMV matrix

Munger

Ong

2014

Journal of Mathematical Physics

View full text Add to dashboard Cite

We prove results about the Hölder continuity of the spectral measures of the extended CMV matrix, given power law bounds of the solution of the eigenvalue equation. We thus arrive at a unitary analogue of the results of Damanik, Killip, and Lenz ["Uniform spectral properties of one-dimensional quasicrystals, III. α-continuity," Commun. Math. Phys. 212, 191-204 (2000)] about the spectral measure of the discrete Schrödinger operator. C 2014 AIP Publishing LLC.

show abstract

On spectral theory for Schrödinger operators with strongly singular potentials

Gesztesy¹,

Zinchenko

2006

Mathematische Nachrichten

Self Cite

168

View full text Add to dashboard Cite

Abstract. We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrödinger operators on [a, ∞), a ∈ R, with a regular finite end point a and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2 × 2 matrix-valued Herglotz functions representing the associated Weyl-Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schrödinger operators on (a, ∞) with a potential strongly singular at the end point a. We focus on situations where the potential is so singular that the associated maximally defined Schrödinger operator is selfadjoint (equivalently, the associated minimally defined Schrödinger operator is essentially self-adjoint) and hence no boundary condition is required at the finite end point a. For this case we show that the Weyl-Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schrödinger operators.

show abstract

Weyl–Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle

Cited by 57 publications

References 34 publications

Equality of the Spectral and Dynamical Definitions of Reflection

Equality of the Spectral and Dynamical Definitions of Reflection

The Hölder continuity of spectral measures of an extended CMV matrix

On spectral theory for Schrödinger operators with strongly singular potentials

Contact Info

Product

Resources

About