Results 11 to 20 of about 147 (25)
Darboux transformations for CMV matrices [PDF]
Advances in Mathematics 298 (2016) 122-206, 2015 We develop a theory of Darboux transformations for CMV matrices, canonical representations of the unitary operators. In perfect analogy with their self-adjoint version -- the Darboux transformations of Jacobi matrices -- they are equivalent to Laurent polynomial modifications of the underlying measures.
arxiv
Borg-Marchenko-type Uniqueness Results for CMV Operators [PDF]
arXiv, 2008 We prove local and global versions of Borg-Marchenko-type uniqueness theorems for half-lattice and full-lattice CMV operators (CMV for Cantero, Moral, and Velazquez \cite{CMV03}). While our half-lattice results are formulated in terms of Weyl-Titchmarsh functions, our full-lattice results involve the diagonal and main off-diagonal Green's functions.
arxiv
The Logic of CMV-Algebras [PDF]
Collana Scientifica dell'Universit\`a di Salerno, pp. 131-151,
Rubbettino Editore, Italy, 2010, 2010 In this paper, once recalled some properties of CMV-algebras, we introduce an expansion of the one-variable fragment of Lukasiewicz propositional logic whose algebraic semantics is the variety of CMV-algebras.
arxiv
A formula related to CMV matrices and Szego cocycles [PDF]
arXiv, 2018 For Schrodinger operators, there is a well known and widely used formula connecting the transfer matrices and Dirichlet determinants. No analog of this formula was previously known for CMV matrices. In this paper we fill this gap and provide the CMV analog of this formula.
arxiv
Cantor Spectrum for CMV and Jacobi Matrices with Coefficients arising from Generalized Skew-Shifts [PDF]
arXiv, 2019 We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$-dense. This implies that the associated CMV and Jacobi matrices have Cantor spectrum for a generic continuous sampling map.
arxiv
Scattering theory for CMV matrices: uniqueness, Helson--Szegő and Strong SzegŐ theorems [PDF]
arXiv, 2010 We develop a scattering theory for CMV matrices, similar to the Faddeev--Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for the uniqueness, which are connected with the Helson--Szeg\H o and the Strong Szeg\H o theorems. The
arxiv
Purely Singular Continuous Spectrum for Limit-Periodic CMV Operators with Applications to Quantum Walks [PDF]
arXiv, 2016 We show that a generic element of a space of limit-periodic CMV operators has zero-measure Cantor spectrum. We also prove a Craig--Simon type theorem for the density of states measure associated with a stochastic family of CMV matrices and use our construction from the first part to prove that the Craig--Simon result is optimal in general.
arxiv
Purely Singular Continuous Spectrum for Sturmian CMV Matrices via Strengthened Gordon Lemmas [PDF]
Proc. Amer. Math. Soc. 145 (2017), 225-239, 2015 The Gordon Lemma refers to a class of results in spectral theory which prove that strong local repetitions in the structure of an operator preclude the existence of eigenvalues for said operator. We expand on recent work of Ong and prove versions of the Gordon Lemma which are valid for CMV matrices and which do not restrict the parity of scales upon ...
arxiv
Faddeev-Marchenko scattering for CMV matrices and the Strong Szego Theorem [PDF]
arXiv, 2008 B. Simon proved the existence of the wave operators for the CMV matrices with Szego class Verblunsky coefficients, and therefore the existence of the scattering function. Generally, there is no hope to restore a CMV matrix when we start from the scattering function, in particular, because it does not contain any information about the (possible ...
arxiv
Finite-gap CMV matrices: Periodic coordinates and a Magic Formula [PDF]
arXiv, 2019 We prove a bijective unitary correspondence between 1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum and 2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class arises as spectrally-dependent operator M\"obius transforms of certain generating CMV matrices which
arxiv