Half-line Schrödinger operators with no bound states

Acta Mathematica

Volumn 193, Issue 1, 2004, Pages 31-72

  Damanik, David ^a   Killip, Rowan ^a  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 12144251598     PISSN: 00015962     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02392550     Document Type: Article

References (34)

1. CARLESON, L., Selected Problems on Exceptional Sets. Van Nostrand Math. Stud., 13. Van Nostrand, Princeton, NJ, 1967.
2. CODDINGTON, E. A. & LEVINSON, N., Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955.
3. CWIKEL, M., Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. of Math., 106 (1977), 93-100.
4. DAMANIK, D., HUNDERTMARK, D., KILLIP, R. & SIMON, B., Variational estimates for discrete Schrödinger operators with potentials of indefinite sign. Comm. Math. Phys., 238 (2003), 545-562.
5. DAMANIK, D., HUNDERTMARK, D. & SIMON, B., Bound states and the Szego condition for Jacobi matrices and Schrödinger operators. J. Funct. Anal., 205 (2003), 357-379.
6. DEIFT, P. & KILLIP, R., On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials. Comm. Math. Phys., 203 (1999), 341-347.
7. DENISOV, S. A., On the application of some of M. G. Krein's results to the spectral analysis of Sturm-Liouville operators. J. Math. Anal. Appl., 261 (2001), 177-191.
8. EASTHAM, M. S. P. & KALF, H., Schrödinger-type Operators with Continuous Spectra. Res. Notes Math., 65. Pitman, Boston, MA, 1982.
9. GERONIMUS, YA. L., Polynomials Orthogonal on a Circle and Their Applications. Amer. Math. Soc. Transl., 104. Amer. Math. Soc., Providence, RI, 1954.
10. GILBERT, D. J. & PEARSON, D. B., On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators. J. Math. Anal. Appl., 128 (1987), 30-56.
11. GOL'DSHEID, I. YA., MOLCHANOV, S. A. & PASTUR, L. A., A random homogeneous Schrödinger operator has a pure point spectrum. Funktsional. Anal. i Prilozhen., 11 (1977), 1-10 (Russian); English translation in Funct. Anal. Appl., 11 (1977), 1-8.
12. GOL'DSHEID, I. YA., MOLCHANOV, S. A. & PASTUR, L. A., A random homogeneous Schrödinger operator has a pure point spectrum. Funktsional. Anal. i Prilozhen., 11 (1977), 1-10 (Russian); English translation in Funct. Anal. Appl., 11 (1977), 1-8.
13. JITOMIRSKAYA, S. & LAST, Y., Power-law subordinacy and singular spectra, I. Half-line operators. Acta Math., 183 (1999), 171-189.
14. KILLIP, R. &: SIMON, B., Sum rules for Jacobi matrices and their applications to spectral theory. Ann. of Math., 158 (2003), 253-321.
15. KISELEV, A., LAST, Y. & SIMON, B., Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators. Comm. Math. Phys., 194 (1998), 1-45.
16. KOTANI, S. & SIMON, B., Localization in general one-dimensional random systems, II. Continuum Schrödinger operators. Comm. Math. Phys., 112 (1987), 103-119.
17. KREǏN, M. G., Continuous analogues of propositions on polynomials orthogonal on the unit circle. Dokl. Akad. Nauk SSSR, 105 (1955), 637-640 (Russian).
18. LIEB, E., Bounds on the eigenvalues of the Laplace and Schrödinger operators. Bull. Amer. Math. Soc., 82 (1976), 751-753.
19. VON NEUMANN, J. & WIGNER, E. P., Über merkwürdige diskrete Eigenwerte. Z. Phys., 30 (1929), 465-467.
20. PEARSON, D. B., Singular continuous measures in scattering theory. Comm. Math. Phys., 60 (1978), 13-36.
21. REMLING, C., The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials. Comm. Math. Phys., 193 (1998), 151-170.
22. ROZENBLYUM, G. V., Distribution of the discrete spectrum of singular differential operators. Dokl. Akad. Nauk SSSR, 202 (1972), 1012-1015 (Russian); English translation in Soviet Math. Dokl., 13 (1972), 245-249.
23. ROZENBLYUM, G. V., Distribution of the discrete spectrum of singular differential operators. Dokl. Akad. Nauk SSSR, 202 (1972), 1012-1015 (Russian); English translation in Soviet Math. Dokl., 13 (1972), 245-249.
24. SCHUR, I., Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, I; II. J. Reine Angew. Math., 147 (1917), 205-232; Ibid., 148 (1918), 122-145 (see also [23]).
25. SCHUR, I., Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, I; II. J. Reine Angew. Math., 147 (1917), 205-232; Ibid., 148 (1918), 122-145 (see also [23]).
26. _ On power series which are bounded in the interior of the unit circle, I; II, in I. Schur Methods in Operator Theory and Signal Processing, pp. 31-59; 61-88. Oper. Theory Adv. Appl., 18. Birkhäuser, Basel, 1986 (English translation of [22]).
27. SIMON, B., The bound state, of weakly coupled Schrödinger operators in one and two dimensions. Ann. Physics, 97 (1976), 279-288.
28. _ Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators. Proc. Amer. Math. Soc., 124 (1996), 3361-3369.
29. _ Orthogonal Polynomials on the Unit Circle. To be published by Amer. Math. Soc.
30. STOLZ, G., Bounded solutions and absolute continuity of Sturm-Liouville operators. J. Math. Anal. Appl., 169 (1992), 210-228.
31. SZEGO, G., Orthogonal Polynomials, 4th edition. Amer. Math. Soc. Colloq. Publ., 23. Amer. Math. Soc., Providence, RI, 1975.
32. TESCHL, G., Jacobi Operators and Completely Integrable Nonlinear Lattices. Math. Surveys Monogr., 72. Amer. Math. Soc., Providence, RI, 2000.
33. WEIDMANN, J., Spectral Theory of Ordinary Differential Operators. Lecture Notes in Math., 1258. Springer-Verlag, Berlin, 1987.
34. ZYGMUND, A., Trigonometric Series, Vols. I and II, 2nd edition. Cambridge Univ. Press, London-New York, 1968.