On New Relations between Spectral Properties of Jacobi Matrices and Their Coefficients

Communications in Mathematical Physics

Volumn 241, Issue 1, 2003, Pages 91-110

  Laptev, A ^a   Naboko, S ^b   Safronov, O ^a  

^a ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

^b ST PETERSBURG STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0142199385     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00220-003-0924-3     Document Type: Article

References (29)

1. Akhiezer, N.I.: The classical moment problem and some related questions in analysis. New York: Hafner Publishing Co., 1965
2. Belov, S., Rybkin, A.: On the existence of WKB-type asymptotics for the generalized eigenvectors of discrete string operators. Preprint
3. Birman, M.S., Krein, M.G.: On the theory of wave operators and scattering operators (Russian). Dokl. Akad. Nauk SSSR 144, 475-478 (1962)
4. Case, K.M.: Orthogonal polynomials from the viewpoint of scattering theory. J. Math. Phys. 16, 2166-2174 (1974)
5. Case, K.M.: Orthogonal Polynomials. II. J. Math. Phys 16, 1435-1440 (1975)
6. Christ, M., Kiselev, A., Remling, C.: The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials. Math. Res. Lett. 4(5), 719-723 (1997)
7. Christ, M., Kiselev, A.: Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results. J. Am. Math. Soc. 11(4), 771-797 (1998)
8. Deift, P., Killip, R.: On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials. Commun. Math. Phys. 203, 341-347 (1998)
9. Denisov, S.A.: On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm-Liouville operators with square summable potential. Preprint
10. Denisov, S.A.: On the existence of the absolutely continuous component for the spectral measure associated with some Krein systems and Sturm-Liouville operators. Commun. Math. Phys. 226, 205-220 (2002)
11. Denisov, S.A.: On the Nevai's Conjecture and Rakhmanov's Theorem for Jacobi Matrices. Private communication
12. Eilenberger, G.: Solitons. Mathematical Methods for Physicists. Springer Series in Solid-State Sciences. 19, Berlin-Heidelberg-New York: Springer-Verlag, 1981
13. Gilbert, D., Pearson, D.B.: On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators. J. Math. Anal. Appl. 128, 30-56 (1987)
14. Janas, J., Naboko, S.: Jacobi matrices with absolutely continuous spectrum. Proc. Am. Math. Soc. 127(3), 791-800 (1999)
15. Khan, S., Pearson, D.B.: Subordinacy and Spectral Theory for finite matrices. Helv. Phys. Acta 65(3), 505-527 (1992)
16. Killip, R.: Perturbations of One-Dimensional Schrödinger operators preserving the absolutely continuous spectrum. Preprint
17. Killip, R., Simon, B.: Sum rules for Jacobi matrices and their application to Spectral Theory. Accepted by Ann. Math.
18. Kiselev, A.: Absolute continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials. Commun. Math. Phys. 179, 377-400 (1996)
19. Kiselev, A., Last, Y., Simon, B.: Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators. Commun. Math. Phys. 194, 1-45 (1998)
20. Krutikov, D., Remling, Ch.: Schrödinger operators with sparse potentials: asymptotics of the Fourier transform of the spectral measure. Preprint
21. Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators. Invent. Math. 135, 329-367 (1999)
22. Molchanov, S., Novitskii, M., Vainberg, B.: First KdV Integrals and Absolutely Continuous Spectrum for 1-D Schrödinger Operator. Commun. Math. Phys. 216, 195-213 (2001)
23. Naboko, S.N.: On the dense point spectrum of Schrödinger and Dirac operators. Teoret. Mat. Fyz. 68(1), 18-28 (1986)
24. Nevai, P.: Orthogonal polynomial, recurrences, Jacobi matrices, and measures. In: Progress in Approximation Theory. (Tampa, FL, 1990), Springer Ser. Comput. Math., 19, New York: Springer, 1992, pp. 79-104
25. Privalov, I.I.: Boundary properties of analytic functions. 2d ed. Moscow-Leningrad: Gosudarstv. Izdat. Tehn.-Teor. Lit., 1950
26. Remling, C.: The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials. Commun. Math. Phys. 193(1), 151-170 (1998)
27. Simon, B.: Some Jacobi matrices with decaying potential and dense point spectrum. Commun. Math. Phys. 87(2), 253-258 (1982)
28. Simon, B.: Some Schrödinger operators with dense point spectrum. Proc. Am. Math. Soc. 125, 203-208 (1997)
29. Yafaev, D.R.: Mathematical scattering theory. Translations of Mathematical Monographs, 105, Providence, RI: American Mathematical Society, 1992