Wave operators on Sobolev spaces

Mizutani, Haruya

doi:10.1090/proc/14838

Wave operators on Sobolev spaces

Author: Haruya Mizutani
Journal: Proc. Amer. Math. Soc. 148 (2020), 1645-1652
MSC (2010): Primary 35P25; Secondary 35Q41, 35Q55
DOI: https://doi.org/10.1090/proc/14838
Published electronically: November 19, 2019
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Abstract: We provide a simple sufficient condition in an abstract framework to deduce the existence and completeness of wave operators (resp., modified wave operators) on Sobolev spaces from the existence and completeness of the usual wave operators (resp., modified wave operators). We then give some examples of Schrödinger operators for which our abstract result applies. An application to scattering theory for the nonlinear Schrödinger equation with a potential is also given.

[1] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable models in quantum mechanics, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005. With an appendix by Pavel Exner. MR 2105735
[2] Valeria Banica and Nicola Visciglia, Scattering for NLS with a delta potential, J. Differential Equations 260 (2016), no. 5, 4410–4439. MR 3437592, https://doi.org/10.1016/j.jde.2015.11.016
[3] Nicolas Burq, Fabrice Planchon, John G. Stalker, and A. Shadi Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J. 53 (2004), no. 6, 1665–1680. MR 2106340, https://doi.org/10.1512/iumj.2004.53.2541
[4] Jan Dereziński and Christian Gérard, Scattering theory of classical and quantum 𝑁-particle systems, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. MR 1459161
[5] B. Helffer and J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger operators (Sønderborg, 1988) Lecture Notes in Phys., vol. 345, Springer, Berlin, 1989, pp. 118–197 (French). MR 1037319, https://doi.org/10.1007/3-540-51783-9_19
[6] Younghun Hong, Scattering for a nonlinear Schrödinger equation with a potential, Commun. Pure Appl. Anal. 15 (2016), no. 5, 1571–1601. MR 3538870, https://doi.org/10.3934/cpaa.2016003
[7] Alexandru D. Ionescu and Wilhelm Schlag, Agmon-Kato-Kuroda theorems for a large class of perturbations, Duke Math. J. 131 (2006), no. 3, 397–440. MR 2219246, https://doi.org/10.1215/S0012-7094-06-13131-9
[8] Tosio Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162 (1965/66), 258–279. MR 190801, https://doi.org/10.1007/BF01360915
[9] David Lafontaine, Scattering for NLS with a potential on the line, Asymptot. Anal. 100 (2016), no. 1-2, 21–39. MR 3570874, https://doi.org/10.3233/ASY-161384
[10] Jing Lu, Changxing Miao, and Jason Murphy, Scattering in 𝐻¹ for the intercritical NLS with an inverse-square potential, J. Differential Equations 264 (2018), no. 5, 3174–3211. MR 3741387, https://doi.org/10.1016/j.jde.2017.11.015
[11] Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR 529429
[12] D. R. Yafaev, Mathematical scattering theory, Translations of Mathematical Monographs, vol. 105, American Mathematical Society, Providence, RI, 1992. General theory; Translated from the Russian by J. R. Schulenberger. MR 1180965