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CMV Matrices with Super Exponentially Decaying VerblunskyCoefficients

Published online by Cambridge University Press:  17 July 2014

M. Zinchenko
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Abstract

We investigate several equivalent notions of the Jost solution associated with a unitaryCMV matrix and provide a necessary and sufficient conditions for the Jost solution toconsist of entire functions of finite growth order in terms of super exponential decay ofVerblunsky coefficients. We also establish several one-to-one correspondences between CMVmatrices with super-exponentially decaying Verblunsky coefficients and spectral dataassociated with the first component of the Jost solution.

Keywords

Jost solutionCMV matrixInverse problem
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 282 - 294
Copyright
© EDP Sciences, 2014

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References

Brown, B. M., Knowles, I., Weikard, R.. On the inverse resonance problem. J. London Math. Soc., 68 (2003), no. 2, 383–401. CrossRefGoogle Scholar
Brown, B. M., Naboko, S., Weikard, R.. The inverse resonance problem for Jacobi operators. Bull. London Math. Soc., 37 (2005), no. 5, 727–737.CrossRefGoogle Scholar
Brown, B. M., Naboko, S., Weikard, R.. The inverse resonance problem for Hermite operators. Constr. Approx., 30 (2009), no. 2, 155–174. CrossRefGoogle Scholar
Bunse-Gerstner, A., Elsner, L.. Schur parameter pencils for the solution of the unitary eigenproblem. Linear Algebra Appl., 154/156 (1991), 741–778. CrossRefGoogle Scholar
Cantero, M. J., Moral, L., Velázquez, L.. Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl., 362 (2003), 29–56. CrossRefGoogle Scholar
D. Damanik, B. Simon. Jost functions and Jost solutions for Jacobi matrices. II. Decay and analyticity. Int. Math. Res. Not. (2006), Art. ID 19396, 1–32.
Faddeyev, L. D.. The inverse problem in the quantum theory of scattering. J. Mathematical Phys., 4 (1963), 72–104. CrossRefGoogle Scholar
Froese, R.. Asymptotic distribution of resonances in one dimension. J. Differential Equations, 137 (1997), no. 2, 251–272. CrossRefGoogle Scholar
Gesztesy, F., Zinchenko, M.. Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle. J. Approx. Theory, 139 (2006), no. 1-2, 172–213. CrossRefGoogle Scholar
L.-S. Hahn, B. Epstein. Classical complex analysis. Jones & Bartlett Learning, 1996.
Hitrik, M.. Bounds on scattering poles in one dimension. Comm. Math. Phys., 208 (1999), no. 2, 381–411. CrossRefGoogle Scholar
Iantchenko, A., Korotyaev, E.. Periodic Jacobi operator with finitely supported perturbation on the half-lattice. Inverse Problems, 27 (2011), no. 11, 115003, 26. CrossRefGoogle Scholar
Iantchenko, A., Korotyaev, E.. Periodic Jacobi operator with finitely supported perturbations: the inverse resonance problem. J. Differential Equations, 252 (2012), no. 3, 2823–2844. CrossRefGoogle Scholar
Iantchenko, A., Korotyaev, E.. Resonances for periodic Jacobi operators with finitely supported perturbations. J. Math. Anal. Appl., 388 (2012), no. 2, 1239–1253. CrossRefGoogle Scholar
Korotyaev, E.. Inverse resonance scattering on the half line. Asymptot. Anal., 37 (2004), no. 3-4, 215–226. Google Scholar
Korotyaev, E.. Stability for inverse resonance problem. Int. Math. Res. Not. (2004), no. 73, 3927–3936. CrossRefGoogle Scholar
Korotyaev, E.. Inverse resonance scattering for Jacobi operators. Russ. J. Math. Phys., 18 (2011), no. 4, 427–439. CrossRefGoogle Scholar
Korotyaev, E.. Resonance theory for perturbed Hill operator. Asymptot. Anal., 74 (2011), no. 3-4, 199–227. Google Scholar
Marletta, M., Naboko, S., Shterenberg, R., Weikard, R.. On the inverse resonance problem for Jacobi operators – uniqueness and stability. J. Anal. Math., 117 (2012), 221–247. CrossRefGoogle Scholar
Marletta, M., Shterenberg, R., Weikard, R.. On the inverse resonance problem for Schrödinger operators. Comm. Math. Phys., 295 (2010), no. 2, 465–484. CrossRefGoogle Scholar
Marletta, M., Weikard, R.. Stability for the inverse resonance problem for a Jacobi operator with complex potential. Inverse Problems, 23 (2007), no. 4, 1677–1688. CrossRefGoogle Scholar
Shterenberg, R., Weikard, R., Zinchenko, M.. Stability for the inverse resonance problem for the CMV operator. Proc. Sympos. Pure Math., 87 (2013), 315–326. CrossRefGoogle Scholar
Simon, B.. Resonances in one dimension and Fredholm determinants. J. Funct. Anal., 178 (2000), no. 2, 396–420. CrossRefGoogle Scholar
Simon, B.. Orthogonal polynomials on the unit circle: new results. Int. Math. Res. Not. (2004), no. 53, 2837–2880. CrossRefGoogle Scholar
Simon, B.. OPUC on one foot. Bull. Amer. Math. Soc. (N.S.), 42 (2005), no. 4, 431–460 (electronic). CrossRefGoogle Scholar
B. Simon. Orthogonal polynomials on the unit circle. Part 1: Classical theory, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005.
B. Simon. Orthogonal polynomials on the unit circle. Part 2: Spectral theory, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005.
Simon, B.. CMV matrices: five years after. J. Comput. Appl. Math., 208 (2007), no. 1, 120–154. CrossRefGoogle Scholar
G. Teschl. Jacobi operators and completely integrable nonlinear lattices. Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000.
Watkins, D. S.. Some perspectives on the eigenvalue problem. SIAM Rev., 35 (1993), no. 3, 430–471. CrossRefGoogle Scholar
Weikard, R., Zinchenko, M.. The inverse resonance problem for CMV operators. Inverse Problems, 26 (2010), no. 5, 55012–55021. CrossRefGoogle Scholar
Zworski, M.. Distribution of poles for scattering on the real line. J. Funct. Anal., 7 (1987), no. 2, 277–296. CrossRefGoogle Scholar
Zworski, M.. A remark on isopolar potentials. SIAM J. Math. Anal., 32 (2001), no. 6, 1324–1326 (electronic). CrossRefGoogle Scholar