In this note, we show how to prove decay estimates for Schrodinger equations by virial methods. The virial method used in this note are based on the series of work by Kowalczyk, Martel, Munoz and Van Den Bosch [10, 11, 12, 13, 14].
[1] Mashael Alammari and Stanley Snelson, Linear and orbital stability analysis for solitary-wave
solutions of variable-coefficient scalar-field equations, J. Hyperbolic Differ. Equ. 19 (2022),
no. 1, 175-201.
[2] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, and H. Holden, Solvable models in quantum
mechanics, second ed., AMS Chelsea Publishing, Providence, RI, 2005, With an appendix by
Pavel Exner.
[3] Shu-Ming Chang, Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai, Spectra of linearized
operators for NLS solitary waves, SIAM J. Math. Anal. 39 (2007/08), no. 4, 1070--1111.
[4] Scipio Cuccagna and Masaya Maeda, On stability of small solitons of the 1-D NLS with a
trapping delta potential, SIAM J. Math. Anal. 51 (2019), no. 6, 4311-4331.
[5] Scipio Cuccagna and Masaya Maeda, On Selection of Standing Wave at Small Energy in the
lD Cubic Schrodinger Equation with a Trapping Potential, Comm. Math. Phys. 396 (2022),
no. 3, 1135-1186.
[6] Scipio Cuccagna and Masaya Maeda, Asymptotic stability of kink with internal modes under
odd perturbation, NoDEA Nonlinear Differential Equations Appl. 30 (2023), no. 1, Paper No.
1, 47.
[7] Scipio Cuccagna, Masaya Maeda, and Stefano Scrobogna, Small energy stabilization for lD
nonlinear Klein Gordon equations, J. Differential Equations 350 (2023), 52-88.
[8] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979),
no. 2, 121-251.
[9] Tosio Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162
(1965/66), 258-279.
[10] Michal Kowalczyk and Yvan Martel, Kink dynamics under odd perturbations for (1+ 1)-scalar
field models with one internal mode, arXiv preprint arXiv:2203.04143.
70
[11] Michal Kowalczyk, Yvan Martel, and Claudio Munoz, Kink dynamics in the ¢ 4 model:
asymptotic stability for odd perturbations in the energy space, J. Amer. Math. Soc. 30 (2017),
no. 3, 769-798.
[12] MichalKowalczyk, Yvan Martel, and Claudio Munoz, Nonexistence of small, odd breathers for
a class of nonlinear wave equations, Lett. Math. Phys. 107 (2017), no. 5, 921-931.
[13] MichalKowalczyk, Yvan Martel, and Claudio Munoz, Soliton dynamics for the lD NLKG
equation with symmetry and in the absence of internal modes, J. Eur. Math. Soc. (JEMS)
24 (2022), no. 6, 2133-2167.
[14] MichalKowalczyk, Yvan Martel, Claudio Munoz, and Hanne Van Den Bosch, A sufficient
condition for asymptotic stability of kinks in general (1 + 1)-scalar field models, Ann. PDE
7 (2021), no. 1, Paper No. 10, 98.
[15] Yongming Li and Jonas Liihrmann, Soliton dynamics for the lD quadratic Klein-Gordon
equation with symmetry, J. Differential Equations 344 (2023), 172-202.
[16] Igor Rodnianski and Jacob Sterbenz, On the formation of singularities in the critical 0(3)
O"-model, Ann. of Math. (2) 172 (2010), no. 1, 187-242.
Masaya Maeda
Department of Mathematics and Informatics, Faculty of Science, Chiba University,
Chiba 263-8522, Japan
E-mail Address: maeda@math. s . chi ba-u. ac . j p
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