Right Limits and Reflectionless Measures for CMV Matrices

Breuer, Jonathan; Ryckman, E.; Zinchenko, Maxim

doi:10.1007/s00220-009-0839-8

Cited by 6 publications

(10 citation statements)

References 38 publications

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“…λ ∈ e and all n, In particular, in (1.17), ⇒ can be replaced by ⇔. However, this is not true for CMV matrices [4].…”

Section: Theorem 11 ([7])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%

“…There is an equivalent definition using Schur functions (see, e.g., [4]). The equivalence is an easy computation using the relations between the Carathéodory and Schur functions (see, e.g., [17]).…”

Section: The CMV Casementioning

confidence: 99%

“…The equivalence is an easy computation using the relations between the Carathéodory and Schur functions (see, e.g., [17]). It is known [17] that (4.8) for one n implies it for all n. It is also known [4] that while (4.8) implies δ n , (C + z)/(C − z)δ n has purely real boundary values a.e. on e, the converse can be false.…”

Section: The CMV Casementioning

confidence: 99%

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Equality of the Spectral and Dynamical Definitions of Reflection

Breuer

Ryckman

Simon

2009

Commun. Math. Phys.

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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.

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“…λ ∈ e and all n, In particular, in (1.17), ⇒ can be replaced by ⇔. However, this is not true for CMV matrices [4].…”

Section: Theorem 11 ([7])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%

Section: The CMV Casementioning

confidence: 99%

Section: The CMV Casementioning

confidence: 99%

See 2 more Smart Citations

Equality of the Spectral and Dynamical Definitions of Reflection

Breuer

Ryckman

Simon

2009

Commun. Math. Phys.

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“…This in turn implies a strong determinism between half-line restrictions of the CMV matrix. The reflectionless property is intimately related to absolute continuity of the spectrum [4,19,28] via the following fundamental results: . If σ(E) is homogeneous and E is reflectionless thereupon, then E has purely a.c. spectrum.…”

Section: Well-approximated Limit-periodic CMV Matrices Are Reflectionmentioning

confidence: 99%

Spectral approximation for ergodic CMV operators with an application to quantum walks

Fillman

Ong

VandenBoom

2018

Journal of Mathematical Analysis and Applications

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We establish concrete criteria for fully supported absolutely continuous spectrum for ergodic CMV matrices and purely absolutely continuous spectrum for limit-periodic CMV matrices. We proceed by proving several variational estimates on the measure of the spectrum and the vanishing set of the Lyapunov exponent for CMV matrices, which represent CMV analogues of results obtained for Schrödinger operators due to Y. Last in the early 1990s. Having done so, we combine those estimates with results from inverse spectral theory to obtain purely absolutely continuous spectrum.

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On the Direct Cauchy Theorem in Widom Domains: Positive and Negative Examples Peter Yuditskii

Yuditskii

2012

Comput. Methods Funct. Theory

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We discuss several questions which remained open in our joint work with M. Sodin "Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions". In particular, we show that there exists a non-homogeneous set E such that the Direct Cauchy Theorem (DCT) holds in the Widom domain C \ E. On the other hand we demonstrate that the weak homogeneity condition on E (introduced recently by Poltoratski and Remling) does not ensure that DCT holds in the corresponding Widom domain. possess the property Re R k,k (x + i0) = 0 for almost all x ∈ E. As usual e k 's denote the standard basis in l 2 . For an exceptional role of this class of Jacobi matrices see [17].The function R(z) = R k,k (z) has positive imaginary part in the upper half plane, and therefore possesses the representationWe follow the terminology in [15] and call reflectionless the measures σ related to reflectionless functions R(z) (1.1), Re R(x + i0) = 0, a.e. x ∈ E. The collection of reflectionless functions associated to the given compact E can be parameterized in the following way. We chose arbitrary x j ∈ [a j , b j ] and set.( 1.2) By D(E) we denote the set of so-called divisors D, where3) The map J(E) → D(E) is defined in the following way. For a reflectionless J the resolvent function R 0,0 (z) possesses the representation (1.2) and this representation produces the collection {x j }. To define ǫ j we represent J as a two-dimensional perturbation of the block-diagonal sum of one-sided Jacobi matrices J ± , that is,This representation generates the identity − 1 R 0,0 (z) = − p 2 0 r − (z) + r + (z), (1.4) where r + (z) = (J + − z) −1 e 0 , e 0 , r − (z) = (

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Right Limits and Reflectionless Measures for CMV Matrices

Cited by 6 publications

References 38 publications

Equality of the Spectral and Dynamical Definitions of Reflection

Equality of the Spectral and Dynamical Definitions of Reflection

Spectral approximation for ergodic CMV operators with an application to quantum walks

On the Direct Cauchy Theorem in Widom Domains: Positive and Negative Examples Peter Yuditskii

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