Finite gap potentials and WKB asymptotics for one-dimensional Schrödinger operators

Communications in Mathematical Physics

Volumn 223, Issue 2, 2001, Pages 409-435

  Kriecherbauer, Thomas ^a   Remling, Christian ^b  

^a UNIVERSITY OF MUNICH   (Germany)

^b UNIVERSITY OF OSNABRÜCK   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035540546     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200100550     Document Type: Article

References (30)

1. 1. Christ, M. and Kiselev, A.: Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results. J. Am. Math. Soc. 11, 771-797 (1998)
2. 2. Christ, M. and Kiselev, A.: WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials. Preprint (2000)
3. 3. Christ, M. and Kiselev, A.: WKB and spectral analysis of one-dimensional Schrödinger operators whose potentials have slowly decaying derivatives. Preprint (2000)
4. 4. Coddington, E. A. and Levinson, N.: Theory of Ordinary Differential Equations. New York: McGraw-Hill, 1955
5. 5. Deift, P. and Killip, R.: On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials. Commun. Math. Phys. 203, 341-347 (1999)
6. 6. Deift, P., Kriecherbauer, T. and Venakides, S.: Forced lattice vibrations II. Commun. Pure Appl. Math. 48, 1251-1298 (1995)
7. 7. Eastham, M.S.P.: The Asymptotic Solution of Linear Differential Systems, London Math. Soc. Monographs New Series 4. Oxford: Clarendon Press, 1989
8. 8. Farkas, H.M. and Kra, I.: Riemann Surfaces, New York: Springer, 1980
9. 9. Gesztesy, F., Ratnaseelan, R. and Teschl, G.: The KdV hierarchy and associated trace formulas. In: Gohberg, I. (ed.) et al., Oper. Theory: Advances and Applications 87, 125-163 (1996)
10. 10. Gesztesy, F. and Weikard, R.: Spectral deformations and soliton equations, In: Ames, W.F., (ed.) et al., Differential equations with applications to mathematical physics, Boston: Academic Press, 1993, pp. 101-139
11. 11. Gilbert, D.J. and Pearson, D.B.: On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators. J. Math. Anal. Appl. 128, 30-56 (1987)
12. 12. Kahane, J.-P.: Some Random Series of Functions. Cambridge: Cambridge University Press, 1985
13. 13. Killip, R.: Perturbations of one-dimensional Schrödinger operators preserving the absolutely continuous spectrum, Ph.D. thesis, Caltech 2000; electronically available at http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=00-326
14. 14. Kiselev, A., Last, Y. and Simon, B.: Modifed Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators. Commun. Math. Phys. 194, 1-45 (1998)
15. 15. Kotani, S. and Ushiroya, N.: One-dimensional Schrödinger operators with random decaying potentials. Commun. Math. Phys. 115, 247-266 (1988)
16. 16. McKean, H.P.: Variation on a theme of Jacobi. Commun. Pure Appl. Math. 38, 669-678 (1985)
17. 17. Mumford, D.: Tata Lectures on Theta 2. Basel: Birkhäuser-Verlag, 1984
18. 18. Naboko, S.N.: Dense point spectra of Schrödinger and Dirac operators. Theor. and Math. Phys. 68, 646-653 (1986)
19. 19. Remling, C.: The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials. Commun. Math. Phys. 193, 151-170 (1998)
20. 20. Remling, C.: Bounds on embedded singular spectrum for one-dimensional Schrödinger operators. Proc. Amer. Math. Soc. 128, 161-171 (2000)
21. 21. Remling, C.: Schrödinger operators with decaying potentials: some counterexamples. Duke Math. J. 105, 463-496 (2000)
22. 22. Rogers, C.A.: Hausdorff Measures. Cambridge: Cambridge University Press, 1970
23. 23. Salem, R. and Zygmund A.: Some properties of trigonometric series whose terms have random signs. Acta Math. 91, 245-301 (1954)
24. 24. Simon, B.: Some Jacobi matrices with decaying potential and dense point spectrum. Commun. Math. Phys. 87, 253-258 (1982)
25. 25. Simon, B.: Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators. Proc. Amer. Math. Soc. 124, 3361-3369 (1996)
26. 26. Simon, B.: Some Schrödinger operators with dense point spectrum. Proc. Amer. Math. Soc. 125, 203-208 (1997)
27. 27. Simon, B.: Schrödinger operators in the twenty-first century. Fokal, A. et al (eds.), Mathematical Physics 2000, London: Imperial College, 2000, pp. 283-288
28. 28. Springer, G.: Introduction to Riemann Surfaces. Reading, MA: Addison-Wesley, 1957
29. 29. Stolz, G.: Bounded solutions and abolute continuity of Sturm-Liouville operators. J. Math. Anal. Appl. 169, 210-228 (1992)
30. 30. von Neumann, J. and Wigner, E.: Über merkwürdige diskrete Eigenwerte. Phys. Z. 30, 465-467 (1929)