Local spectral properties of reflectionless Jacobi, CMV, and Schrödinger operators. (English) Zbl 1165.34051
The authors deal with spectral properties of self-adjoint Jacobi and Schrödinger operators \(H\) on \(\mathbb{Z}\) and \(\mathbb{R}\), respectively, and unitary CMV operators \(U\) on \(\mathbb{Z}\), which are reflections on a homogeneous set \(\mathcal{E}\) contained in the essential spectrum. They prove that under the assumption of a Blaschke - type condition on their discrete spectra accumulating at \(\mathcal{E}\), the operators \(H\) (resp. \(U\)) have purely absolutely continuous spectrum on \(\mathcal{E}\).
Reviewer: Petru A. Cojuhari (Kraków)
MSC:
| 34L05 | General spectral theory of ordinary differential operators |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 47A10 | Spectrum, resolvent |
References:
| [1] | Akhiezer, N. I., The Classical Moment Problem (1965), Oliver & Boyd: Oliver & Boyd Edinburgh · Zbl 0135.33803 |
| [2] | Akhiezer, N. I.; Glazman, I. M., Theory of Operators in Hilbert Space, vol. I (1981), Pitman: Pitman Boston · Zbl 0467.47001 |
| [3] | Aronszajn, N.; Donoghue, W. F., On exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. Anal. Math., 5, 321-388 (1956/57) · Zbl 0138.29502 |
| [4] | Avron, J.; Simon, B., Almost periodic Schrödinger operators, I. Limit periodic potentials, Comm. Math. Phys., 82, 101-120 (1981) · Zbl 0484.35069 |
| [5] | Belokolos, E. D.; Bobenko, A. I.; Enol’skii, V. Z.; Its, A. R.; Matveev, V. B., Algebro-Geometric Approach to Nonlinear Integrable Equations (1994), Springer: Springer Berlin · Zbl 0809.35001 |
| [6] | Carleson, L., On \(H^\infty\) in multiply connected domains, (Beckner, W.; Calderón, A. P.; Fefferman, R.; Jones, P. W., Conference on Harmonic Analysis in Honor of Antoni Zygmund, vol. II (1983), Wadsworth: Wadsworth CA), 349-372 · Zbl 0527.46048 |
| [7] | Carmona, R.; Lacroix, J., Spectral Theory of Random Schrödinger Operators (1990), Birkhäuser: Birkhäuser Basel · Zbl 0717.60074 |
| [8] | J.S. Christiansen, B. Simon, M. Zinchenko, Finite gap Jacobi matrices, I. The isospectral torus, in preparation; J.S. Christiansen, B. Simon, M. Zinchenko, Finite gap Jacobi matrices, I. The isospectral torus, in preparation · Zbl 1200.42012 |
| [9] | Chulaevskii, V. A., Inverse spectral problem for limit-periodic Schrödinger operators, Funct. Anal. Appl., 18, 230-233 (1984) · Zbl 0599.34028 |
| [10] | Craig, W., The trace formula for Schrödinger operators on the line, Comm. Math. Phys., 126, 379-407 (1989) · Zbl 0681.34026 |
| [11] | De Concini, C.; Johnson, R. A., The algebraic-geometric AKNS potentials, Ergodic Theory Dynam. Systems, 7, 1-24 (1987) · Zbl 0636.35077 |
| [12] | Deift, P.; Simon, B., Almost periodic Schrödinger operators, III. The absolutely continuous spectrum in one dimension, Comm. Math. Phys., 90, 389-411 (1983) · Zbl 0562.35026 |
| [13] | Donoghue, W. F., Monotone Matrix Functions and Analytic Continuation (1974), Springer: Springer Berlin · Zbl 0278.30004 |
| [14] | Dubrovin, B. A.; Matveev, V. B.; Novikov, S. P., Non-linear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties, Russian Math. Surveys, 31, 1, 59-146 (1976) · Zbl 0346.35025 |
| [15] | Egorova, I. E., On a class of almost periodic solutions of the KdV equation with a nowhere dense spectrum, Russian Acad. Sci. Dokl. Math., 45, 290-293 (1990) |
| [16] | Fisher, S., Function Theory on Planar Domains (1983), John Wiley & Sons: John Wiley & Sons New York · Zbl 0511.30022 |
| [17] | Ford, L. R., Automorphic Functions (1951), Chelsea: Chelsea New York · Zbl 1364.30001 |
| [18] | Gesztesy, F.; Holden, H., Soliton Equations and Their Algebro-Geometric Solutions. Vol. I: \((1 + 1)\)-Dimensional Continuous Models, Cambridge Stud. Adv. Math., vol. 79 (2003), Cambridge Univ. Press · Zbl 1061.37056 |
| [19] | Gesztesy, F.; Krishna, M.; Teschl, G., On isospectral sets of Jacobi operators, Comm. Math. Phys., 181, 631-645 (1996) · Zbl 0881.58069 |
| [20] | F. Gesztesy, K.A. Makarov, M. Zinchenko, Local AC spectrum for reflectionless Jacobi, CMV, and Schrödinger operators, Acta Appl. Math., in press; F. Gesztesy, K.A. Makarov, M. Zinchenko, Local AC spectrum for reflectionless Jacobi, CMV, and Schrödinger operators, Acta Appl. Math., in press |
| [21] | Gesztesy, F.; Simon, B., The \(ξ\) function, Acta Math., 176, 49-71 (1996) · Zbl 0885.34070 |
| [22] | Gesztesy, F.; Yuditskii, P., Spectral properties of a class of reflectionless Schrödinger operators, J. Funct. Anal., 241, 486-527 (2006) · Zbl 1387.34120 |
| [23] | 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 |
| [24] | 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 |
| [25] | D.J. Gilbert, Subordinacy and spectral analysis of Schrödinger operators, PhD thesis, University of Hull, 1984; D.J. Gilbert, Subordinacy and spectral analysis of Schrödinger operators, PhD thesis, University of Hull, 1984 |
| [26] | Gilbert, D. J., On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints, Proc. Roy. Soc. Edinburgh Sect. A, 112, 213-229 (1989) · Zbl 0678.34024 |
| [27] | Gilbert, D. J., On subordinacy and spectral multiplicity for a class of singular differential operators, Proc. Roy. Soc. Edinburgh Sect. A, 128, 549-584 (1998) · Zbl 0909.34022 |
| [28] | Gilbert, D. J.; Pearson, D. B., On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl., 128, 30-56 (1987) · Zbl 0666.34023 |
| [29] | Johnson, R. A., The recurrent Hill’s equation, J. Differential Equations, 46, 165-193 (1982) · Zbl 0535.34021 |
| [30] | Johnson, R. A., On the Sato-Segal-Wilson solutions of the K-dV equation, Pacific J. Math., 132, 343-355 (1988) · Zbl 0662.35093 |
| [31] | Johnson, R.; Moser, J., The rotation number for almost periodic potentials, Comm. Math. Phys., 84, 403-438 (1982) · Zbl 0497.35026 |
| [32] | Jones, P. W.; Marshall, D. E., Critical points of Green’s function, harmonic measure, and the corona problem, Ark. Mat., 23, 281-314 (1985) · Zbl 0589.30028 |
| [33] | Kac, I. S., On the multiplicity of the spectrum of a second-order differential operator, Sov. Math. Dokl., 3, 1035-1039 (1962) · Zbl 0131.31201 |
| [34] | Kac, I. S., Spectral multiplicity of a second order differential operator and expansion in eigenfunctions, Izv. Akad. Nauk SSSR. Izv. Akad. Nauk SSSR, Izv. Akad. Nauk SSSR, 28, 951-952 (1964), (in Russian) · Zbl 0238.34034 |
| [35] | Kac, I. S.; Krein, M. G., \(R\)-functions—analytic functions mapping the upper halfplane into itself, (Amer. Math. Soc. Transl. Ser. 2, vol. 103 (1974)), 1-18 · Zbl 0291.34016 |
| [36] | Koosis, P., Introduction to \(H_p\) Spaces, Cambridge Tracts in Math. (1998), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 1024.30001 |
| [37] | Kotani, S., Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators, (Itô, K., Stochastic Analysis (1984), North-Holland: North-Holland Amsterdam), 225-247 · Zbl 0549.60058 |
| [38] | Kotani, S., One-dimensional random Schrödinger operators and Herglotz functions, (Itô, K.; Ikeda, N., Probabilistic Methods in Mathematical Physics (1987), Academic Press: Academic Press New York), 219-250 · Zbl 0647.60073 |
| [39] | Kotani, S., Link between periodic potentials and random potentials in one-dimensional Schrödinger operators, (Knowles, I. W., Differential Equations and Mathematical Physics (1987), Springer: Springer Berlin), 256-269 · Zbl 0653.60050 |
| [40] | Kotani, S.; Krishna, M., Almost periodicity of some random potentials, J. Funct. Anal., 78, 390-405 (1988) · Zbl 0644.60061 |
| [41] | Levitan, B. M., Almost periodicity of infinite-zone potentials, Math. USSR Izv., 18, 249-273 (1982) · Zbl 0483.35075 |
| [42] | Levitan, B. M., Approximation of infinite-zone potentials by finite-zone potentials, Math. USSR Izv., 20, 55-87 (1983) · Zbl 0517.34037 |
| [43] | Levitan, B. M., An inverse problem for the Sturm-Liouville operator in the case of finite-zone and infinite-zone potentials, Trans. Moscow Math. Soc., 45, 1, 1-34 (1984) · Zbl 0576.34021 |
| [44] | Levitan, B. M., On the closure of the set of finite-zone potentials, Math. USSR Sb., 51, 67-89 (1985) · Zbl 0589.34026 |
| [45] | Levitan, B. M., Inverse Sturm-Liouville Problems (1987), VNU Science Press: VNU Science Press Utrecht · Zbl 0749.34001 |
| [46] | Levitan, B. M.; Savin, A. V., Examples of Schrödinger operators with almost periodic potentials and nowhere dense absolutely continuous spectrum, Sov. Math. Dokl., 29, 541-544 (1984) · Zbl 0586.34019 |
| [47] | Marchenko, V. A., Sturm-Liouville Operators and Applications (1986), Birkhäuser: Birkhäuser Basel · Zbl 0592.34011 |
| [48] | Moser, J., An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum, Comment. Math. Helv., 56, 198-224 (1981) · Zbl 0477.34018 |
| [49] | F. Nazarov, A. Volberg, P. Yuditskii, Reflectionless measures with singular components, preprint, arXiv: 0711.0948; F. Nazarov, A. Volberg, P. Yuditskii, Reflectionless measures with singular components, preprint, arXiv: 0711.0948 |
| [50] | Novikov, S.; Manakov, S. V.; Pitaevskii, L. P.; Zakharov, V. E., Theory of Solitons (1984), Consultants Bureau: Consultants Bureau New York · Zbl 0598.35002 |
| [51] | Pastur, L.; Figotin, A., Spectra of Random and Almost-Periodic Operators (1992), Springer: Springer Berlin · Zbl 0752.47002 |
| [52] | Pastur, L. A.; Tkachenko, V. A., Spectral theory of a class of one-dimensional Schrödinger operators with limit-periodic potentials, Trans. Moscow Math. Soc., 115-166 (1989) · Zbl 0709.34064 |
| [53] | Peherstorfer, F.; Yuditskii, P., Asymptotic behavior of polynomials orthonormal on a homogeneous set, J. Anal. Math., 89, 113-154 (2003) · Zbl 1032.42028 |
| [54] | Peherstorfer, F.; Yuditskii, P., Almost periodic Verblunsky coefficients and reproducing kernels on Riemann surfaces, J. Approx. Theory, 139, 91-106 (2006) · Zbl 1103.30007 |
| [55] | Popovici, I.; Volberg, A., Boundary Harnack principle for Denjoy domains, Complex Var. Theory Appl., 37, 471-490 (1998) · Zbl 1054.30508 |
| [56] | Priwalow, I. I., Randeigenschaften analytischer Funktionen (1956), VEB Verlag: VEB Verlag Berlin · Zbl 0073.06501 |
| [57] | Ransford, T., Potential Theory in the Complex Plane (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0828.31001 |
| [58] | C. Remling, The absolutely continuous spectrum of Jacobi matrices, arXiv: 0706.1101; C. Remling, The absolutely continuous spectrum of Jacobi matrices, arXiv: 0706.1101 · Zbl 1235.47032 |
| [59] | Remling, C., The absolutely continuous spectrum of one-dimensional Schrödinger operators, Math. Phys. Anal. Geom., 10, 359-373 (2007) · Zbl 1139.81036 |
| [60] | Rosenblum, M.; Rovnyak, J., Topics in Hardy Classes and Univalent Functions (1994), Birkhäuser: Birkhäuser Basel · Zbl 0816.30001 |
| [61] | Simon, B., Analogs of the \(m\)-function in the theory of orthogonal polynomials on the unit circle, J. Comput. Appl. Math., 171, 411-424 (2004) · Zbl 1055.33010 |
| [62] | Simon, B., Orthogonal polynomials on the unit circle: New results, Int. Math. Res. Not., 53, 2837-2880 (2004) · Zbl 1086.42501 |
| [63] | Simon, B., Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, Part 2: Spectral Theory, Amer. Math. Soc. Colloq. Publ., vol. 54 (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1082.42020 |
| [64] | Simon, B., OPUC on one foot, Bull. Amer. Math. Soc., 42, 431-460 (2005) · Zbl 1108.42005 |
| [65] | Simon, B., On a theorem of Kac and Gilbert, J. Funct. Anal., 223, 109-115 (2005) · Zbl 1071.47027 |
| [66] | Simon, B., CMV matrices: Five years later, J. Comput. Appl. Math., 208, 120-154 (2007) · Zbl 1125.15027 |
| [67] | Simon, B., Equilibrium measures and capacities in spectral theory, Inverse Probl. Imaging, 1, 713-772 (2007) · Zbl 1149.31004 |
| [68] | B. Simon, Szegő’s Theorem and Its Descendants: Spectral Theory for \(L^2\); B. Simon, Szegő’s Theorem and Its Descendants: Spectral Theory for \(L^2\) |
| [69] | Sims, R., Reflectionless Sturm-Liouville equations, J. Comput. Appl. Math., 208, 207-225 (2007) · Zbl 1132.34021 |
| [70] | Sodin, M.; Yuditskii, P., Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum and pseudoextendible Weyl functions, Russian Acad. Sci. Dokl. Math., 50, 512-515 (1995) · Zbl 0868.34063 |
| [71] | Sodin, M.; Yuditskii, P., Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum, Comment. Math. Helv., 70, 639-658 (1995) · Zbl 0846.34024 |
| [72] | Sodin, M.; Yuditskii, P., Almost-periodic Sturm-Liouville operators with homogeneous spectrum, (Boutel de Monvel, A.; Marchenko, A., Algebraic and Geometric Methods in Mathematical Physics (1996), Kluwer), 455-462 · Zbl 0844.34089 |
| [73] | 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, 387-435 (1997) · Zbl 1041.47502 |
| [74] | 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 |
| [75] | Zinsmeister, M., Espaces de Hardy et domaines de Denjoy, Ark. Mat., 27, 363-378 (1989) · Zbl 0682.30030 |
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