Dispersion estimates for discrete Schrödinger and Klein–Gordon equations
HTML articles powered by AMS MathViewer
- by
E. A. Kopylova
Translated by: the author - St. Petersburg Math. J. 21 (2010), 743-760
- DOI: https://doi.org/10.1090/S1061-0022-2010-01115-4
- Published electronically: July 14, 2010
- PDF | Request permission
Abstract:
The long-time asymptotics is derived for solutions of the discrete $3$-dimensional Schrödinger and Klein–Gordon equations.References
- P. M. Bleher, Operators that depend meromorphically on a parameter, Vestnik Moskov. Univ. Ser. I Mat. Meh. 24 (1969), no. 5, 30–36 (Russian, with English summary). MR 0265956
- B. R. Vaĭnberg, The short-wave asymptotic behavior of the solutions of stationary problems, and the asymptotic behavior as $t\rightarrow \infty$ of the solutions of nonstationary problems, Uspehi Mat. Nauk 30 (1975), no. 2(182), 3–55 (Russian). MR 0415085
- B. R. Vaĭnberg, Asimptoticheskie metody v uravneniyakh matematicheskoĭ fiziki, Moskov. Gos. Univ., Moscow, 1982 (Russian). MR 700559
- I. M. Glazman, Pryamye metody kachestvennogo spektral′nogo analiza, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0185471
- H. Islami and B. Vainberg, Large time behavior of solutions to difference wave operators, Comm. Partial Differential Equations 31 (2006), no. 1-3, 397–416. MR 2209760, DOI 10.1080/03605300500361529
- Arne Jensen and Tosio Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J. 46 (1979), no. 3, 583–611. MR 544248
- Arne Jensen and Gheorghe Nenciu, A unified approach to resolvent expansions at thresholds, Rev. Math. Phys. 13 (2001), no. 6, 717–754. MR 1841744, DOI 10.1142/S0129055X01000843
- A. I. Komech, E. A. Kopylova, and M. Kunze, Dispersive estimates for 1D discrete Schrödinger and Klein-Gordon equations, Appl. Anal. 85 (2006), no. 12, 1487–1508. MR 2282998, DOI 10.1080/00036810601074321
- A. I. Komech, E. A. Kopylova, and B. R. Vainberg, On dispersive properties of discrete 2D Schrödinger and Klein-Gordon equations, J. Funct. Anal. 254 (2008), no. 8, 2227–2254. MR 2402107, DOI 10.1016/j.jfa.2008.01.005
- Peter D. Lax and Ralph S. Phillips, Scattering theory, 2nd ed., Pure and Applied Mathematics, vol. 26, Academic Press, Inc., Boston, MA, 1989. With appendices by Cathleen S. Morawetz and Georg Schmidt. MR 1037774
- Minoru Murata, Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal. 49 (1982), no. 1, 10–56. MR 680855, DOI 10.1016/0022-1236(82)90084-2
- W. Schlag, Dispersive estimates for Schrödinger operators: a survey, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 255–285. MR 2333215
- W. Shaban and B. Vainberg, Radiation conditions for the difference Schrödinger operators, Appl. Anal. 80 (2001), no. 3-4, 525–556. MR 1914696, DOI 10.1080/00036810108841007
- M. S. Èskina, The scattering problem for partial-difference equations, Math. Phys. No. 3 (1967) (Russian), “Naukova Dumka”, Kiev, 1967, pp. 248–273 (Russian, with English summary). MR 0211027
Bibliographic Information
- E. A. Kopylova
- Affiliation: Institute for Information Transmission Problems, Russian Academy of Sciences, B. Karetnyi 19, Moscow 101447, Russia
- Email: ek@vpti.vladimir.ru
- Received by editor(s): November 19, 2009
- Published electronically: July 14, 2010
- Additional Notes: Supported partly by FWF (grant no. P19138-N13), DFG (grant no. 436 RUS 113/929/0-1), and RFBR (grants nos. 07-01-00018a and 06-01-00096)
- © Copyright 2010 American Mathematical Society
- Journal: St. Petersburg Math. J. 21 (2010), 743-760
- MSC (2010): Primary 39A14, 35L10
- DOI: https://doi.org/10.1090/S1061-0022-2010-01115-4
- MathSciNet review: 2604564