Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials

Geometric and Functional Analysis

Volumn 12, Issue 6, 2002, Pages 1174-1234

  Christ, Michael ^a   Kiselev, Alexander ^b,c  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF CHICAGO   (United States)

^c UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036991106     PISSN: 1016443X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00039-002-1174-9     Document Type: Article

References (44)

1. [AK] P.K. ALSHOLM, T. KATO, Scattering with long range potentials, Partial Diff. Eq., Proc. Symp. Pure Math. 23, Amer. Math. Soc., Providence, Rhode Island (1973), 393-399.
2. [ArD] N. ARONSZAJN, W. DONOGHUE, On exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. Analyse Math. 15 (1957), 321-388.
3. [B] M. BEN-ARTZI, On the absolute continuity of Schrödinger operators with spherically symmetric, long-range potentials. I, II, J. Differential Equations 38 (1980), 41-50, 51-60.
4. [BD] M. BEN-ARTZI, A. DEVINATZ, Spectral and scattering theory for the adiabatic oscillator and related potentials, J. Math. Phys. 20 (1979), 594-607.
5. [BeS] F. BEREZIN, M. SHUBIN, The Schrödinger Equation, Mathematics and its Applications, Kluwer Academic Publishers, Soviet series 66, 1991.
6. [BiS] M. BIRMAN, M. SOLOMYAK, Spectral Theory of Selfadjoint Operators in Hubert Space, D. Reidel Publishing, Dordrecht, 1987.
7. [BuM] V.S. BUSLAEV, V.B. MATVEEV, Wave operators for Schrödinger equation with slowly decreasing potentials, Theor. Math. Phys. 2 (1970), 266-274.
8. [CK1] M. CHRIST, A. KISELEV, Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results, J. Amer. Math. Soc. 11 (1998), 771-797.
9. [CK2] M. CHRIST, A. KISELEV, WKB asymptotics of generalized eigenfunctions of one-dimensional Schrödinger oeprators, J. Funct.; Anal. 179 (2001), 426-447.
10. [CK3] M. CHRIST, A. KISELEV, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), 409-425.
11. [CK4] M. CHRIST, A. KISELEV, WKB and spectral analysis of one-dimensional Schrödinger operators with slowly varying potentials, Comm. Math. Phys. 218 (2001), 245-262.
12. [CK5] M. CHRIST, A. KISELEV, One-dimensional Schrödinger operators with slowly decaying potentials: Spectra and Asymptotics, Institute for Pure and Applied Mathematics Lecture Notes, March 2001.
13. [CoL] E.A. CODDINGTON, N. LEVINSON, Theory of Ordinary Differential Equations. MCGraw-Hill, New York, 1955.
14. [CyFKS] H. CYCON, R. FROESE, W. KIRSCH, B. SIMON, Schrödinger Operators, Springer-Verlag, 1987.
15. [DK] P. DEIFT, R. KILLIP, On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Commun. Math. Phys. 203 (1999), 341-347.
16. [De] S.A. DENISOV, On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm-Liouville operators with square summable potential, preprint.
17. [DeK] S. DENISOV, S. KUPIN, On the singular spectrum of Schrodinger operators with decaying potentials, preprint.
18. [Dev] A. DEVINATZ, The existence of wave operators for oscillating potentials, J. Math. Phys. 21 (1980), 2406-2411.
19. [Dol] J.D. DOLLARD, Asymptotic convergence and Coulomb interaction, J. Math. Phys. 5 (1964), 729-738.
20. [Do2] J.D. DOLLARD, Quantum mechanical scattering theory for short-range and Coulomb interactions, Rocky Mountain J. Math. 1 (1971), 5-88.
21. [E] M.S.P. EASTHAM, The Asymptotic Solution of Linear Differential Systems, Clarendon Press, Oxford, 1989.
22. [H] L. HÖRMANDER, The existence of wave operators in scattering theory, Math. Z. 146 (1976), 69-91.
23. [K] A. KISELEV, Imbedded singular continuous spectrum for Schrödinger operators, preprint.
24. [KL] A. KISELEV, Y. LAST, Solutions, spectrum and dynamics for Schrödinger operators on infinite domains, Duke Math. J. 102 (2000), 125-150.
25. [KLS] A. KISELEV, Y. LAST, B. SIMON, Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators, Commun. Math. Phys. 194 (1998), 1-45.
26. [KRS] A. KISELEV, C. REMLING, B. SIMON, Effective perturbation methods for one-dimensional Schrödinger operators, J. Differential Equations 151:2 (1999), 290-312.
27. [KoU] S. KOTANI, N. USHIROYA, One-dimensional Schrödinger operators with random decaying potentials, Commun. Math. Phys. 115 (1988), 247-266.
28. [KrR] T. KRIECHERBAUER, C. REMLING, Finite gap potentials and WKB asymptotics for one-dimensional Schrödinger operators, Comm. Math. Phys. 223:2 (2001), 409-435.
29. [N] S.N. NABOKO, Dense point spectra of Schrödinger and Dirac operators, Theor.-math. 68 (1986), 18-28.
30. [NeW] J. VON NEUMANN, E.P. WIGNER, Über merkwürdige diskrete eigenwerte, Z. Phys. 30(1929), 465-467.
31. [NoMPZ] S. NOVIKOV, S.V. MANAKOV, L.P. PITAEVSKII, V.E. ZAKHAROV, Theory of Solitons. The Inverse Scattering Method, (translated from the Russian), Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York-London, 1984.
32. [P] D. PEARSON, Singular continuous measures in scattering theory, Comm. Math. Phys. 60 (1978), 13-36.
33. [RS1] M. REED, B. SIMON, Methods of Modern Mathematical Physics, I. Functional Analysis, Academic Press, New York, 1972.
34. [RS2] M. REED, B. SIMON, Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-adjointness, Academic Press, New York, 1972.
35. [RS3] M. REED, B. SIMON, Methods of Modern Mathematical Physics, III. Scattering Theory, Academic Press, London-San Diego, 1979.
36. [Re1] C. REMLING, The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials, Comm. Math. Phys. 193 (1998), 151-170.
37. [Re2] C. REMLING, Bounds on embedded singular spectrum for one-dimensional Schrödinger operators, Proc. Amer. Math. Soc. 128:1 (2000), 161-171.
38. [Re3] C. REMLING, Schrödinger operators with decaying potentials: some counterexamples, Duke Math. J. 105:3 (2000), 463-496.
39. [RiJLS] R. DEL RIO, S. JITOMIRSKAYA, Y. LAST, B. SIMON, Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69 (1996), 153-200.
40. [RoS] I. RODNIANSKI, W. SCHLAG, Classical and quantum scattering for a class of long range random potentials, preprint.
41. [S1] B. SIMON, Some Schrödinger operators with dense point spectrum, Proc. Amer. Math. Soc. 125 (1997), 203-208.
42. [S2] B. SIMON, Spectral analysis of rank one perturbations and applications, Proc. Mathematical Quantum Theory, II: Schrödinger Operators (J. Feldman, R. Roese, L. Rosen, eds.), CRN Proc. Lecture Notes 8 (1995), 109-149.
43. [T] E.C. TITCHMARSH, Eigenfunction Expansions, 2nd ed., Oxford University Press, Oxford, 1962.
44. [W] D.A.W. WHITE, Schrödinger operators with rapidly oscillating central potentials, Trans. Amer. Math. Soc. 275 (1983), 641-677.