Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

Journal d'Analyse Mathematique

Volumn 99, Issue , 2006, Pages 199-249

  Erdoǧan, M Burak ^a   Schlag, Wilhelm ^b  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b UNIVERSITY OF CHICAGO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33847747008     PISSN: 00217670     EISSN: 15658538     Source Type: Journal    
DOI: 10.1007/BF02789446     Document Type: Article

References (50)

1. S. Agmon, Spectral properties of Schrodinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), 151-218.
2. V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, translated from the French by A. Avez, W. A. Benjamin, Inc., New York - Amsterdam, 1968.
3. H. Berestycki and P. -L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313-345.
4. H. Berestycki and P. -L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal. 82 (1983), 347-375.
5. V. S. Buslaev, and G. S. Perelman, Scattering for the nonlinear Schrodinger equation: states that are close to a soliton, (Russian) Algebra i Analiz 4 (1992), no. 6, 63-102;
6. translation in St. Petersburg Math. J. 4 (1993), 1111-1142.
7. V. S. Buslaev, and G. S. Perelman, On the stability of solitary waves for nonlinear Schrodinger equations, in Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, 1995, 75-98.
8. T. Cazenave and P. -L. Lions, Orbital stability of standing waves for some nonlinear Schrodinger equations, Comm. Math. Phys. 85 (1982), 549-561
9. 3 =0 and a variational characterization of other solutions, Arch. Rat. Mech. Anal. 46 (1972), 81-95.
10. A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math. 56 (2003), 1565-1607.
11. S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 54 (2001), 1110-1145.
12. S. Cuccagna and D. Pelinovsky, Bifurcations from the endpoints of the essential spectrum in the linearized nonlinear Schrödinger problem, J. Math. Phys. 46 (2005), 053520, 15 pp.
13. S. Cuccagna, D. Pelinovsky and V. Vougalter, Spectra of positive and negative energies in the linearized NLS problem, Comm. Pure Appl. Math. 58 (2005), 1-29.
14. S. Denissov, private communication.
15. M. B. Erdoǧan and W. Schlag, Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I, Dynamics of PDE, vol. 1, (2004), 359-379.
16. F. Gesztesy, C. K. R. T. Jones, Y. Latushkin and M. Stanislavova, A spectral mapping theorem and invariant manifolds for nonlinear Schrodinger equations, Indiana Univ. Math. J. 49 (2000), 221-243.
17. M. Goldberg, Dispersive bounds for the three-dimensional Schrodinger equation with almost critical potentials, Geom. Funct. Anal., to appear.
18. M. Goldberg and W. Schlag, Dispersive estimates for Schrodinger operators in dimensions one and three, Comm. Math. Phys. 251, (2004), 157-178.
19. M. Grillakis, Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system, Comm. Pure Appl. Math. 41 (1988), 747-774.
20. M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), 160-197.
21. M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal. 94 (1990), 308-348.
22. P. D. Hislop, and I. M. Sigal, Introduction to Spectral theory. With applications to Schrodinger operators, Springer-Verlag, New York, 1996.
23. A. Jensen and T. Kato, Spectral properties of Schrodinger operators and time-decay of the wave functions, Duke Math. J. 46 (1979), 583-611.
24. A. Jensen, and G. Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys. 13 (2001), 717-754.
25. J. -L. Journé, A. Soffer and C. D. Sogge, Decay estimates for Schrodinger operators, Comm. Pure Appl. Math. 44 (1991), 573-604.
26. n, Arch. Rat. Mech. Anal. 65 (1989), 243-266.
27. R. S. MacKay, Stability of equilibria of Hamiltonian systems, in Nonlinear phenomena and chaos, (Malvern, 1985), Malvern Phys. Sen, Hilger, Bristol, 1986, pp. 254-270.
28. K. McLeod and J. Serrin, Nonlinear Schrodinger equation. Uniqueness of positive solutions of Δu + f(u) = 0 in ℝ, Arch. Rat. Mech. Anal. 99 (1987), 115-145.
29. M. Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal. 49 (1982), 10-56.
30. G. Perelman, On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré 2 (2001), 605-673.
31. G. Perelman, Asymptotic Stability of multi-soliton solutions for nonlinear Schrodinger equations, Comm. Partial Differential Equations 29 (2004), 1051-1095.
32. J. Rauch, Local decay of scattering solutions to Schrödinger's equation, Comm. Math. Phys. 61(1978), 149-168.
33. M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV, Academic Press, New York-London, 1979.
34. I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (2004), 451-513.
35. I. Rodnianski, W. Schlag and A. Soffer, Dispersive analysis of charge transfer models, Comm. Pure Appl. Math. 58 (2005), 149-216.
36. I. Rodnianski,W. Schlag and A. Soffer, Asymptotic stability of N-soliton states of NLS, preprint 2003.
37. W. Schlag, Stable manifolds for an orbitally unstable NLS, Annals of Math., to appear.
38. W. Schlag, Dispersive estimates for Schrodinger operators: A survey. Conference Proceedings "Workshop on Aspects of Non-Linear PDE", IAS Princeton, 2004, to appear.
39. J. Shatah, Stable standing waves of nonlinear Klein-Cordon equations, Comm. Math. Phys. 91(1983), 313-327.
40. J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985), 173-190.
41. A. Soffer and M. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys. 133 (1990), 119-146.
42. A. Soffer and M. Weinstein, Multichannel nonlinear scattering, II. The case of anysotropic potentials and data, J. Diff. Eq. 98 (1992), 376-390.
43. E. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, 1970.
44. W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149-162.
45. W. A. Strauss, Nonlinear Wave Equations, CBMS Regional Conference Series in Mathematics 73, American Mathematical Society, Providence, RI, 1989.
46. C. Sulem and P. -L. Sulem, The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse, Springer-Verlag, New York, 1999.
47. M.I. Weinstein, Modulational stability of ground states of nonlinear Schrodinger equations, SIAM J. Math. Anal. 16 (1985), 472-491.
48. M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), 51-67.
49. kp-continuity of wave operators for Schrodinger operators, J.Math. Soc. Japan 47 (1995), 551-581.
50. K. Yajima, Dispersive estimate for Schrödinger equations with threshold resonance and eigenvalue, Comm. Math. Phys. 259 (2005), 475-509.