Estimates on modulation spaces for Schrödinger operators with time-dependent sub-linear vector potentials. (English) Zbl 1533.35083
Summary: In this paper, we give estimates of the solutions to Schrödinger equation on modulation spaces with vector potential of sub-linear growth.
MSC:
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35K10 | Second-order parabolic equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
References:
| [1] | Bényi A., Gröchenig K., Okoudjou K. A. and Rogers L. G., Unimodular Fourier multipliers for modulation spaces. J. Functional Anal. 246 (2007), 366-384. · Zbl 1120.42010 |
| [2] | Bhimani D. G., Manna R., Nicola F., Thangavelu S. and Trapasso S. I., Phase space analysis of the Hermite semigroup and applications to nonlin-ear global well-posedness. Adv. Math. 392 (2021), Paper No. 107995, 18. · Zbl 1476.35297 |
| [3] | Cordero E., Gröchenig K., Nicola F. and Rodino L., Wiener algebras of Fourier integral operators. J. Math. Pures. Appl. 99 (2013), 219-233. · Zbl 1306.42045 |
| [4] | Cordero E., Nicola F. and Rodino L., Schrödinger equations with rough Hamiltonians. Discrete Contin. Dyn. Syst. 35 (2015), 4805-4821. · Zbl 1334.35458 |
| [5] | Córdoba A. and Fefferman C., Wave packets and Fourier integral operators. Comm. Partial Differential Equations 3 (1978), 979-1005. · Zbl 0389.35046 |
| [6] | Delort J. M., F.B.I. transformation, Lecture Notes in Mathematics, 1522, Springer-Verlag, Berlin (1992). · Zbl 0760.35004 |
| [7] | Feichtinger H. G., Modulation Spaces on Locally Compact Abelian Groups (TECHNICAL REPORT), Universität Wien, 1983. |
| [8] | Folland G. B., Harmonic analysis in phase space, Annals of Mathematics Studies, 122, Princeton University Press, Princeton, NJ, 1989. · Zbl 0682.43001 |
| [9] | Fujiwara D., A construction of the fundamental solution for the Schrödinger equations. Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 10-14. · Zbl 0421.35018 |
| [10] | Gröchenig K., Foundations of Time-Frequency Analysis, Birkhäuser, 2001. · Zbl 0966.42020 |
| [11] | Hörmander L., Fourier integral operators. I. Acta Math. 127 (1971), 79-183. · Zbl 0212.46601 |
| [12] | Iwabuchi T., Navier-Stokes equations and nonlinear heat equations in mod-ulation spaces with negative derivative indices. J. Differential Equations 248 (2010), 1972-2002. · Zbl 1185.35166 |
| [13] | Kato K., Kobayashi M. and Ito S., Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials. J. Functional Anal. 266 (2014), 733-753. · Zbl 1294.35010 |
| [14] | Kato K., Kobayashi M. and Ito S., Representation of Schrödinger opera-tor of a free particle via short-time Fourier transform and its applications. Tohoku Math. J. 64 (2012), 223-231. · Zbl 1246.35171 |
| [15] | Kato T., Remarks on Schrödinger operators with vector potentials. Integral Equations Operator Theory 1 (1978), 103-113. · Zbl 0395.47023 |
| [16] | Kobayashi M. and Sugimoto M., The inclusion relation between Sobolev and modulation spaces. J. Functional Anal. 260 (2011), 3189-3208. · Zbl 1232.46033 |
| [17] | Leinfelder H. and Simader C. G., Schrödinger operators with singular mag-netic vector potentials. Math. Z. 176 (1981), 1-19. · Zbl 0468.35038 |
| [18] | Muramatsu R., Estimates on modulation spaces for Schrödinger operators with first order magnetic fields. SUT Journal of Mathematics 57 (2021), 201-209. · Zbl 1486.35161 |
| [19] | Nicola F., Phase space analysis of semilinear parabolic equations. J. Funct. Anal. 267 (2014), 727-743. · Zbl 1296.35225 |
| [20] | Simon B., Schrödinger operators with singular magnetic vector potentials. Math. Z. 131 (1973), 361-370. · Zbl 0277.47006 |
| [21] | Staffilani G. and Tataru D., Strichartz estimates for a Schrödinger opera-tor with nonsmooth coefficients. Comm. Partial Differential Equations 27 (2002), 1337-1372. · Zbl 1010.35015 |
| [22] | Tataru D., Phase space transforms and microlocal analysis (Phase space analysis of partial differential equations). Pubbl. Cent. Ric. Mat. Ennio Giorgi, II, 505-524, Scuola Norm. Sup., Pisa, 2004. · Zbl 1111.35143 |
| [23] | Tataru D., Parametrices and dispersive estimates for Schrödinger operators with variable coefficients. Amer. J. Math. 130 (2008), 571-634. · Zbl 1159.35315 |
| [24] | Wang B. and Hudzik H., The global Cauchy problem for the NLS and NLKG with small rough data. J. Differential Equations 232 (2007), 36-73. · Zbl 1121.35132 |
| [25] | Wang B., Zhao L. and Guo B., Isometric decomposition operators, function spaces E λ p,q and applications to nonlinear evolution equations. J. Functional Anal. 233 (2006), 1-39. · Zbl 1099.46023 |
| [26] | Yajima K., Schrödinger evolution equations with magnetic fields. J. Analyse Math. 56 (1991), 29-76. · Zbl 0739.35083 |
| [27] | Zworski M., Semiclassical analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, Providence, RI, 2012. · Zbl 1252.58001 |
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