$\mathcal {L}^p$ boundedness of the scattering wave operators of Schrödinger dynamics with time-dependent potentials and applications–part I
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- by Avy Soffer and Xiaoxu Wu;
- Trans. Amer. Math. Soc. 378 (2025), 4437-4507
- DOI: https://doi.org/10.1090/tran/9414
- Published electronically: March 25, 2025
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Abstract:
This paper establishes the $\mathcal {L}^p$ boundedness of wave operators localized at high-frequency for linear Schrödinger equations in $\mathbb {R}^3$ with time-dependent potentials. The approach to the proof is based on new cancellation lemmas. As a typical application based on this method, combined with Strichartz estimates is the existence and scattering for nonlinear dispersive equations. For example, we prove global existence and uniform boundedness in $\mathcal {L}^{\infty }$, for a class of Hartree nonlinear Schrödinger equations in $\mathcal {L}^2(\mathbb {R}^3)$, allowing the presence of solitons. We also prove the existence of free channel wave operators in $\mathcal {L}^p(\mathbb {R}^3)$ for $p>6$.References
- Marius Beceanu, New estimates for a time-dependent Schrödinger equation, Duke Math. J. 159 (2011), no. 3, 417–477. MR 2831875, DOI 10.1215/00127094-1433394
- J. Bourgain, Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential, Comm. Math. Phys. 204 (1999), no. 1, 207–247. MR 1705671, DOI 10.1007/s002200050644
- Nicolas Burq, Fabrice Planchon, John G. Stalker, and A. Shadi Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal. 203 (2003), no. 2, 519–549. MR 2003358, DOI 10.1016/S0022-1236(03)00238-6
- 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, DOI 10.1512/iumj.2004.53.2541
- M. Beceanu and A. Soffer, The Schrödinger equation with potential in random motion, Preprint, arXiv:1111.4584, 2011.
- Marius Beceanu and Avy Soffer, The Schrödinger equation with a potential in rough motion, Comm. Partial Differential Equations 37 (2012), no. 6, 969–1000. MR 2924464, DOI 10.1080/03605302.2012.668257
- M. Beceanu and Soffer, A semilinear Schröedinger equation with random potential, Preprint, arXiv:1903.03451, 2019.
- Marius Beceanu and Wilhelm Schlag, Structure formulas for wave operators under a small scaling invariant condition, J. Spectr. Theory 9 (2019), no. 3, 967–990. MR 4003547, DOI 10.4171/JST/268
- K. Cai, Fine properties of charge transfer models, Preprint, math-ph/0311048, 2003.
- M. Christ, J. Colliander, and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, Preprint, math/0311048, 2003.
- G. Chen, Strichartz estimates for wave equations with charge transfer Hamiltonian, Preprint, arXiv:1610.05226, 2016.
- Gong Chen and Jacek Jendrej, Asymptotic stability and classification of multi-solitons for Klein-Gordon equations, Comm. Math. Phys. 405 (2024), no. 1, Paper No. 7, 47. MR 4691857, DOI 10.1007/s00220-023-04904-5
- Michael Christ and Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409–425. MR 1809116, DOI 10.1006/jfan.2000.3687
- O. Costin, J. L.Lebowitz, and A. Rokhlenko, Ionization of a model atom: exact results and connection with experiment, arXiv preprint physics/9905038, 1999.
- E. B. Davies, Time-dependent scattering theory, Math. Ann. 210 (1974), 149–162. MR 348530, DOI 10.1007/BF01360037
- Volker Enss and Krešimir Veselić, Bound states and propagating states for time-dependent Hamiltonians, Ann. Inst. H. Poincaré Sect. A (N.S.) 39 (1983), no. 2, 159–191. MR 722684
- K. O. Friedrichs, On the perturbation of continuous spectra, Communications on Appl. Math. 1 (1948), 361–406. MR 29100, DOI 10.1002/cpa.3160010404
- M. Goldberg, Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials, Geom. Funct. Anal. 16 (2006), no. 3, 517–536. MR 2238943, DOI 10.1007/s00039-006-0568-5
- Michael Goldberg, Strichartz estimates for the Schrödinger equation with time-periodic $L^{n/2}$ potentials, J. Funct. Anal. 256 (2009), no. 3, 718–746. MR 2484934, DOI 10.1016/j.jfa.2008.11.005
- L. Grafakos, Classical Fourier analysis, vol. 2, Springer, 2008.
- A. Galtbayar, A. Jensen, and K. Yajima, Local time-decay of solutions to Schrödinger equations with time-periodic potentials, J. Statist. Phys. 116 (2004), no. 1-4, 231–282. MR 2083143, DOI 10.1023/B:JOSS.0000037203.79298.ec
- J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 144 (1992), no. 1, 163–188. MR 1151250, DOI 10.1007/BF02099195
- J. S. Howland, Born series and scattering by time-dependent potentials, Rocky Mountain J. Math. (1980), 521–531.
- James S. Howland, Stationary scattering theory for time-dependent Hamiltonians, Math. Ann. 207 (1974), 315–335. MR 346559, DOI 10.1007/BF01351346
- Arne Jensen, Spectral properties of Schrödinger operators and time-decay of the wave functions results in $L^{2}(\textbf {R}^{m})$, $m\geq 5$, Duke Math. J. 47 (1980), no. 1, 57–80. MR 563367
- Arne Jensen, Spectral properties of Schrödinger operators and time-decay of the wave functions. Results in $L^{2}(\textbf {R}^{4})$, J. Math. Anal. Appl. 101 (1984), no. 2, 397–422. MR 748579, DOI 10.1016/0022-247X(84)90110-0
- 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
- J. M. Jauch, R. Lavine, and R. G. Newton, Scattering into cones, Helv. Phys. Acta 45 (1972/73), 325–330. MR 418733
- J.-L. Journé, A. Soffer, and C. D. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math. 44 (1991), no. 5, 573–604. MR 1105875, DOI 10.1002/cpa.3160440504
- Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048, DOI 10.1353/ajm.1998.0039
- O. C. J. Lebowitz and S. Tanveer, Ionization of Coulomb systems in $\mathbb {R}^3$ by time periodic forcings of arbitrary size, 2006.
- C. Møller, General properties of the characteristic matrix in the theory of elementary particles. 1, Matematisk-fysiske Meddelelser Kongelige Danske Videnskabernes Selskab, vol. 23, issue 1, 1945, pp. 1–48.
- Frank Merle, Pierre Raphaël, and Jeremie Szeftel, On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation, Duke Math. J. 163 (2014), no. 2, 369–431. MR 3161317, DOI 10.1215/00127094-2430477
- F. Nier and A. Soffer, Dispersion and Strichartz estimates for some finite rank perturbations of the Laplace operator, J. Funct. Anal. 198 (2003), no. 2, 511–535. MR 1964550, DOI 10.1016/S0022-1236(02)00099-X
- Galina Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Comm. Partial Differential Equations 29 (2004), no. 7-8, 1051–1095. MR 2097576, DOI 10.1081/PDE-200033754
- Jeffrey Rauch, Local decay of scattering solutions to Schrödinger’s equation, Comm. Math. Phys. 61 (1978), no. 2, 149–168. MR 495958, DOI 10.1007/BF01609491
- M. Reed and B. Simon, Functional analysis, Revised and enlarged ed., 1980.
- M. Reed and B. Simon, Scattering theory, Academic Press, 1979.
- Igor Rodnianski and Wilhelm Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (2004), no. 3, 451–513. MR 2038194, DOI 10.1007/s00222-003-0325-4
- Igor Rodnianski, Wilhelm Schlag, and Avraham Soffer, Dispersive analysis of charge transfer models, Comm. Pure Appl. Math. 58 (2005), no. 2, 149–216. MR 2094850, DOI 10.1002/cpa.20066
- Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
- W. Schlag, Intertwining wave operators, Fourier restriction, and Wiener theorems, Preprint, arXiv:1802.01982, 2018.
- Hart F. Smith and Christopher D. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations 25 (2000), no. 11-12, 2171–2183. MR 1789924, DOI 10.1080/03605300008821581
- A. Soffer and M. I. Weinstein, Nonautonomous Hamiltonians, J. Statist. Phys. 93 (1998), no. 1-2, 359–391. MR 1656374, DOI 10.1023/B:JOSS.0000026738.52652.6e
- A. Soffer and M. I. Weinstein, Ionization and scattering for short-lived potentials, Lett. Math. Phys. 48 (1999), no. 4, 339–352. MR 1709044, DOI 10.1023/A:1007695606961
- A. Soffer and X. Wu, $L^p$ boundedness of the scattering wave operators of Schrödinger dynamics–part 2, Preprint, arXiv:2202.03307, 2022.
- Avy Soffer and Wu Xiaoxu, On the large time asymptotics of Schrödinger type equations with general data, arXiv preprint arXiv:2203.00724, 2022.
- Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
- Terence Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations 25 (2000), no. 7-8, 1471–1485. MR 1765155, DOI 10.1080/03605300008821556
- Terence Tao, Monica Visan, and Xiaoyi Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1281–1343. MR 2354495, DOI 10.1080/03605300701588805
- W.-M. Wang, Logarithmic bounds on Sobolev norms for time dependent linear Schrödinger equations, Comm. Partial Differential Equations 33 (2008), no. 10-12, 2164–2179. MR 2475334, DOI 10.1080/03605300802537115
- Ricardo Weder, $L^p$-$L^{\dot p}$ estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential, J. Funct. Anal. 170 (2000), no. 1, 37–68. MR 1736195, DOI 10.1006/jfan.1999.3507
- Kenji Yajima, $L^p$-boundedness of wave operators for two-dimensional Schrödinger operators, Comm. Math. Phys. 208 (1999), no. 1, 125–152. MR 1729881, DOI 10.1007/s002200050751
- K. Yajima, Dispersive estimates for Schrödinger equations with threshold resonance and eigenvalue, Comm. Math. Phys. 259 (2005), no. 2, 475–509. MR 2172692, DOI 10.1007/s00220-005-1375-9
- Kenji Yajima, The $W^{k,p}$-continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan 47 (1995), no. 3, 551–581. MR 1331331, DOI 10.2969/jmsj/04730551
- K. Yajima, The $L^p$ boundedness of wave operators for Schrödinger operators with threshold singularities I. The odd dimensional case, J. Math. Sci. Univ. Tokyo 13, no. 1, 43.
- Kenji Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), no. 3, 415–426. MR 891945, DOI 10.1007/BF01212420
Bibliographic Information
- Avy Soffer
- Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
- MR Author ID: 198594
- Email: soffer@math.rutgers.edu
- Xiaoxu Wu
- Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
- ORCID: 0000-0003-2972-8153
- Email: xw292@math.rutgers.edu
- Received by editor(s): March 4, 2022
- Received by editor(s) in revised form: January 20, 2023, and December 16, 2024
- Published electronically: March 25, 2025
- Additional Notes: The first author was supported in part by NSF grant DMS-1600749 and by NSFC11671163
The second author is supported by ARC-FL220100072 - © Copyright 2025 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 378 (2025), 4437-4507
- MSC (2020): Primary 35P25, 35Q55, 47A40
- DOI: https://doi.org/10.1090/tran/9414