CMV matrices with asymptotically constant coefficients. Szegő–Blaschke class, scattering theory. (English) Zbl 1167.47029
In this paper, scattering theory is extended for CMV matrices with asymptotically constant Verblunsky coefficients. It is shown that the class of two-sided CMV matrices of the Szegő–Blaschke class correspond precisely to the class for which the scattering problem can be posed and solved. CMV matrices are represented as multiplication operators in \(L^{2}\)-spaces with respect to specific bases. A CMV matrix of this type is determined uniquely by the scattering data, and the associated Gelfand–Levitan–Marchenko transformation operators are bounded.
Reviewer: Petru A. Cojuhari (Kraków)
MSC:
| 47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
| 47A40 | Scattering theory of linear operators |
Keywords:
CMV matrix; Jacobi matrix; scattering theory; \(A_{2}\) condition; Carleson condition; Schur functions; Verblunsky coefficientsReferences:
| [1] | Arov, D.; Dym, H., On matricial Nehari problems, \(J\)-inner matrix functions and the Muckenhoupt condition, J. Funct. Anal., 181, 227-299 (2001) · Zbl 0980.47014 |
| [2] | Arov, D.; Dym, H., Criteria for the strong regularity of \(J\)-inner functions and \(γ\)-generating matrices, J. Math. Anal. Appl., 280, 387-399 (2003) · Zbl 1044.47012 |
| [3] | Beals, R.; Deift, P.; Tomei, C., Direct and Inverse Scattering on the Line, Math. Surveys Monogr., vol. 28 (1988), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0699.34016 |
| [4] | Case, K.; Geronimo, J., Scattering theory and polynomials orthogonal on the real line, Trans. Amer. Math. Soc., 258, 467-494 (1980) · Zbl 0436.42018 |
| [5] | Christ, M.; Kiselev, A., Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials, Geom. Funct. Anal., 12, 1174-1234 (2002) · Zbl 1039.34076 |
| [6] | D. Damanik, R. Killip, B. Simon, Perturbation of orthogonal polynomials with periodic recursion coefficients, preprint, 2006; D. Damanik, R. Killip, B. Simon, Perturbation of orthogonal polynomials with periodic recursion coefficients, preprint, 2006 · Zbl 1194.47031 |
| [7] | Deift, P. A.; Killip, R., On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Comm. Math. Phys., 203, 341-347 (1999) · Zbl 0934.34075 |
| [8] | Denisov, S. A., On Rakhmanov’s theorem for Jacobi matrices, Proc. Amer. Math. Soc., 132, 847-852 (2004) · Zbl 1050.47024 |
| [9] | Denisov, S. A., On the existence of wave operators for some Dirac operators with square summable potential, Geom. Funct. Anal., 14, 3, 529-534 (2004) · Zbl 1070.47504 |
| [10] | Egorova, I.; Michor, J.; Teschl, G., Scattering theory for Jacobi operators with quasi-periodic background, Comm. Math. Phys., 264, 811-842 (2006) · Zbl 1115.39025 |
| [11] | Egorova, I.; Michor, J.; Teschl, G., Scattering theory for Jacobi operators with steplike quasi-periodic background, Inverse Problems, 23, 905-918 (2007) · Zbl 1152.47023 |
| [12] | Garnett, John B., Bounded Analytic Functions, Grad. Texts in Math., vol. 236 (2007), Springer: Springer New York, xiv+459 pp · Zbl 1106.30001 |
| [13] | Gesztesy, F.; Zinchenko, M., Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle, J. Approx. Theory, 139, 172-213 (2006) · Zbl 1118.47023 |
| [14] | Gesztesy, F.; Zinchenko, M., A Borg-type theorem associated with orthogonal polynomials on the unit circle, J. London Math. Soc., 74, 757-777 (2006) · Zbl 1131.47027 |
| [15] | Guseinov, G., The determination of an infinite Jacobi matrix from the scattering data, Soviet Math. Dokl., 17, 596-600 (1976) · Zbl 0336.47021 |
| [16] | Kheifets, A.; Yuditskii, P., An analysis and extension of V.P. Potapov’s approach to interpolation problems, (Matrix and Operator Valued Functions. Matrix and Operator Valued Functions, Oper. Theory Adv. Appl., vol. 72 (1994), Birkhäuser: Birkhäuser Basel), 133-161 · Zbl 0837.47012 |
| [17] | Kheifets, A.; Peherstorfer, F.; Yuditskii, P., On scattering for CMV matrices (2007) |
| [18] | Killip, R.; Simon, B., Sum rules for Jacobi matrices and their applications to spectral theory, Ann. of Math. (2), 158, 253-321 (2003) · Zbl 1050.47025 |
| [19] | Kupin, S.; Peherstorfer, F.; Volberg, A.; Yuditskii, P., Inverse scattering problem for a special class of canonical systems and non-linear Fourier integral. Part I: Asymptotics of eigenfunctions, (Oper. Theory Adv. Appl., vol. 186 (2008), Birkhäuser Verlag: Birkhäuser Verlag Basel), 285-323 · Zbl 1169.34057 |
| [20] | N. Makarov, A. Poltoratski, Beurling-Malliavin theory for Toeplitz kernels, preprint, 2007, arXiv: math/0702497; N. Makarov, A. Poltoratski, Beurling-Malliavin theory for Toeplitz kernels, preprint, 2007, arXiv: math/0702497 · Zbl 1186.47025 |
| [21] | Marchenko, V., Sturm-Liouville Operators and Applications (1986), Birkhäuser: Birkhäuser Basel · Zbl 0592.34011 |
| [22] | Marchenko, V., Nonlinear Equations and Operator Algebras, Math. Appl. (Soviet Ser.), vol. 17 (1988), D. Reidel Publishing Co.: D. Reidel Publishing Co. Dordrecht-Boston, MA · Zbl 0644.47053 |
| [23] | van Moerbekke, P.; Mumford, D., The spectrum of difference operators and algebraic curves, Acta Math., 143, 1-2, 93-154 (1979) · Zbl 0502.58032 |
| [24] | Nikolskii, N., Treatise on the Shift Operator (1986), Springer-Verlag: Springer-Verlag Berlin · Zbl 0587.47036 |
| [25] | Peherstorfer, F.; Yuditskii, P., Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points, Proc. Amer. Math. Soc., 129, 3213-3230 (2001) · Zbl 0976.42012 |
| [26] | Peherstorfer, F.; Yuditskii, P., Asymptotic behavior of polynomials orthonormal on a homogeneous set, J. Anal. Math., 89, 113-154 (2003) · Zbl 1032.42028 |
| [27] | Peherstorfer, F.; Yuditskii, P., Finite difference operators with a finite-band spectrum, (Oper. Theory Adv. Appl., vol. 186 (2008), Birkhäuser Verlag: Birkhäuser Verlag Basel), 345-387 · Zbl 1169.47020 |
| [28] | Privalov, I. I., Graničnye Svoĭstva Analitičeskih Funkciĭ (1950), Gosudarstv. Izdat. Tehn.-Teor. Lit.: Gosudarstv. Izdat. Tehn.-Teor. Lit. Moscow-Leningrad, 336 pp. (in Russian) |
| [29] | Ch. Remling, The absolutely continuous spectrum of Jacobi matrices, preprint, 2007, arXiv: 0706.1101; Ch. Remling, The absolutely continuous spectrum of Jacobi matrices, preprint, 2007, arXiv: 0706.1101 |
| [30] | Simon, B., Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, Amer. Math. Soc. Colloq. Publ. (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1082.42020 |
| [31] | Simon, B., Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, Amer. Math. Soc. Colloq. Publ. (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1082.42021 |
| [32] | Simon, B., A canonical factorization for meromorphic Herglotz functions on the unit disc and sum rules for Jacobi matrices, J. Funct. Anal., 214, 396-409 (2004) · Zbl 1064.30030 |
| [33] | Simon, B., CMV matrices: Five years after, J. Comput. Appl. Math., 208, 1, 120-154 (2007) · Zbl 1125.15027 |
| [34] | Sodin, M.; Yuditskii, P., Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal., 7, 3, 387-435 (1997) · Zbl 1041.47502 |
| [35] | T. Tao, Ch. Thiele, Nonlinear Fourier Analysis, IAS/Park City Mathematics Series, Graduate Summer School 2003, in press; T. Tao, Ch. Thiele, Nonlinear Fourier Analysis, IAS/Park City Mathematics Series, Graduate Summer School 2003, in press |
| [36] | Teschl, G., Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surveys Monogr., vol. 72 (2000), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1056.39029 |
| [37] | Volberg, A.; Yuditskii, P., On the inverse scattering problem for Jacobi matrices with the spectrum on an interval, a finite system of intervals or a Cantor set of positive length, Comm. Math. Phys., 226, 567-605 (2002) · Zbl 1019.34082 |
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