Electromagnetic Schrödinger flow: multiplier methods for dispersion
Fanelli, Luca
Journées équations aux dérivées partielles, (2010), p. 1-13 / Harvested from Numdam

We show a list of results which have been recently obtained about dispersive properties of the electromagnetic Schrödinger flow. We introduce a general philosophy, based on multiplier technique, which permits to detect the bad parts of an electromagnetic potential which can possibly affect the dispersion.

Publié le : 2010-01-01
DOI : https://doi.org/10.5802/jedp.64
@article{JEDP_2010____A7_0,
     author = {Fanelli, Luca},
     title = {Electromagnetic Schr\"odinger flow: multiplier methods for dispersion},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {2010},
     pages = {1-13},
     doi = {10.5802/jedp.64},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_2010____A7_0}
}
Fanelli, Luca. Electromagnetic Schrödinger flow: multiplier methods for dispersion. Journées équations aux dérivées partielles,  (2010), pp. 1-13. doi : 10.5802/jedp.64. http://gdmltest.u-ga.fr/item/JEDP_2010____A7_0/

[1] Barceló, J.A., Ruiz, A., and Vega, L., Some dispersive estimates for Schrödinger equations with repulsive potentials J. Funct. Anal. 236 (2006), 1–24. | MR 2227127 | Zbl pre05037252

[2] Burq, N., Planchon, F., Stalker, J., and Tahvildar-Zadeh, S. Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J. 53(6) (2004), 1665–1680. | MR 2106340 | Zbl 1084.35014

[3] Constantin, P., and Saut, J.-C., Local smoothing properties of dispersive equations, Journ. AMS (1988), 413–439. | MR 928265 | Zbl 0667.35061

[4] Cycon, H.L., Froese, R., Kirsch, W., and Simon, B., Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry Texts and Monographs in Physics, Springer Verlag Berlin Heidelberg New York (1987). | MR 883643 | Zbl 0619.47005

[5] Erdoğan, M. B., Goldberg, M., and Schlag, W., Strichartz and Smoothing Estimates for Schrödinger Operators with Almost Critical Magnetic Potentials in Three and Higher Dimensions, to appear on Forum Math. | MR 2541480 | Zbl 1181.35208

[6] Erdoğan, M. B., Goldberg, M., and Schlag, W., Strichartz and smoothing estimates for Schrodinger operators with large magnetic potentials in 3 , to appear on J. European Math. Soc. | MR 2390334 | Zbl 1152.35021

[7] D’Ancona, P., and Fanelli, L., Strichartz and smoothing estimates for dispersive equations with magnetic potentials, Comm. Part. Diff. Eqns. 33 (2008), 1082–1112. | MR 2424390 | Zbl 1160.35363

[8] P. D’Ancona, and L. Fanelli: Smoothing estimates for the Schrödinger equation with unbounded potentials, Journ. Diff. Eq. 246 (2009), 4552–4567. | MR 2523293 | Zbl 1173.35031

[9] P. D’Ancona, L. Fanelli, L. Vega, and N. Visciglia: Endpoint Strichartz estimates for the magnetic Schrödinger equation, J. Funct. Anal. 258 (2010), 3227–3240. | MR 2601614 | Zbl 1188.81061

[10] Fanelli, L., Non-trapping magnetic fields and Morrey-Campanato estimates for Schrödinger operators, textitJ. Math. Anal. Appl. 357 (2009), 1–14. | MR 2526801 | Zbl 1170.35374

[11] L. Fanelli, and A. García: Counterexamples to Strichartz estimates for the magnetic Schrödinger equation, to appear on Comm. Cont. Math.

[12] Fanelli, L., and Vega, L., Magnetic virial identities, weak dispersion and Strichartz inequalities, to appear on Math. Ann. | MR 2664570 | Zbl 1163.35005

[13] Georgiev, V., Stefanov, A., and Tarulli, M. Smoothing - Strichartz estimates for the Schrödinger equation with small magnetic potential, Discrete Contin. Dyn. Syst. A 17 (2007), 771–786. | MR 2276474 | Zbl 1125.35077

[14] Ginibre, J., and Velo, G., Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995) no. 1, 50–68. | MR 1351643 | Zbl 0849.35064

[15] Goldberg, M., Dispersive estimates for the three-dimensional schrödinger equation with rough potential, Amer. J. Math. 128 (2006), 731–750. | MR 2230923 | Zbl 1096.35027

[16] Goldberg, M., and Schlag, W., Dispersive estimates for schrödinger operators in dimensions one and three, Comm. Math. Phys. 251 (2004), 157–178. | MR 2096737 | Zbl 1086.81077

[17] Goldberg, M., Vega, L., and Visciglia, N., Counterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials, Int. Math Res Not., 2006 Vol. 2006: article ID 13927. | MR 2211154 | Zbl 1102.35026

[18] Ionescu, A.D., and Kenig, C., Well-posedness and local smoothing of solutions of Schrödinger equations, Math. Res. Letters 12 (2005), 193–205. | MR 2150876 | Zbl 1077.35108

[19] Kato, T., and Yajima, K., Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), 481–496. | MR 1061120 | Zbl 0833.47005

[20] Keel, M., and Tao, T., Endpoint Strichartz estimates, Amer. J. Math. 120 (1998) no. 5, 955–980. | MR 1646048 | Zbl 0922.35028

[21] Leinfelder, H., and Simader, C., Schrödinger operators with singular magnetic vector potentials, Math Z. 176 (1981), 1–19. | MR 606167 | Zbl 0468.35038

[22] C.S. Morawetz, Time decay for the nonlinear Klein-Gordon equation, Proc. Roy. Soc. London A, 306 (1968), 291–296. | MR 234136 | Zbl 0157.41502

[23] B. Perthame, and L. Vega, Morrey-Campanato estimates for the Helmholtz Equations, J. Func. Anal. 164 (1999), 340–355. | MR 1695559 | Zbl 0932.35048

[24] Robbiano, L., and Zuily, C., Strichartz estimates for Schrödinger equations with variable coefficients, Mem. Soc. Math. Fr. 101-102 (2005). | Numdam | MR 2193021 | Zbl 1097.35002

[25] Rodnianski, I., and Schlag, W., Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155(3) (2004), 451–513. | MR 2038194 | Zbl 1063.35035

[26] Ruiz, A., and Vega, L., On local regularity of Schr¨odinger equations. Int. Math. Research Notices 1, 1993, 13–27 . | MR 1201747 | Zbl 0812.35016

[27] Sjölin, P., Regularity of solutions to the Schrödinger equations, Duke Math. J. 55 (1987), 699–715. | MR 904948 | Zbl 0631.42010

[28] Stein, E., Harmonic Analysis. Princeton University Press, Princeton, New Jersey, 1993. | MR 1232192 | Zbl 0821.42001

[29] Strichartz, R., Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705–774. | MR 512086 | Zbl 0372.35001

[30] Vega, L., The Schrödinger equation: pointwise convergence to the initial date, Proc. AMS 102 (1988), 874–878. | MR 934859 | Zbl 0654.42014