Abstract
We prove the sharp \({L^1-L^\infty}\) time-decay estimate for the 2D -Schrödinger equation with a general family of scaling critical electromagnetic potentials.
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Barceló J.A., Ruiz A., Vega L.: Weighted estimates for the Helmholtz equation and some applications. J. Funct. Anal. 150, 356–382 (1997)
Beceanu M., Goldberg M.: Schödinger dispersive estimates for a scaling-critical class of potentials. Commun. Math. Phys. 314, 471–481 (2012)
Borg G.: Umkerhrung der Sturm-Liouvillischen Eigebnvertanfgabe Bestimung der difúerentialgleichung die Eigenverte. Acta Math. 78, 1–96 (1946)
Burq N., Planchon F., Stalker J., Tahvildar-Zadeh S.: Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential. J. Funct. Anal. 203(2), 519–549 (2003)
Burq N., Planchon F., Stalker J., Tahvildar-Zadeh S.: Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Indiana Univ. Math. J. 53(6), 1665–1680 (2004)
Cazenave, T.: Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences. American Mathematical Society, New York, Providence (2003)
Erdogan M.B., Goldberg M., Schlag W.: Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions. Forum Math. 21, 687–722 (2009)
Erdogan M.B., Goldberg M., Schlag W.: Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in \({{\mathbb{R}}^3}\). J. Eur. Math. Soc. 10, 507–531 (2008)
D’Ancona P., Fanelli L.: L p-boundedness of the wave operator for the one dimensional Schrödinger operators. Commun. Math. Phys. 268, 415–438 (2006)
D’Ancona P., Fanelli L.: Decay estimates for the wave and Dirac equations with a magnetic potential. Commun. Pure Appl. Math. 60, 357–392 (2007)
D’Ancona P., Fanelli L.: Strichartz and smoothing estimates for dispersive equations with magnetic potentials. Commun. Partial Differ. Equ. 33, 1082–1112 (2008)
D’Ancona P., Fanelli L., Vega L., Visciglia N.: Endpoint Strichartz estimates for the magnetic Schrödinger equation. J. Funct. Anal. 258, 3227–3240 (2010)
Duyckaerts T.: Inégalités de résolvante pour l’opérateur de Schrödinger avec potentiel multipolaire critique. Bull. de la Société mathématique de France 134, 201–239 (2006)
Fanelli L., Felli V., Fontelos M., Primo A.: Time decay of scaling critical electromagnetic Schrödinger flows. Commun. Math. Phys. 324, 1033–1067 (2013)
Fanelli, L., Felli, V., Fontelos, M., Primo, A.: Time decay of scaling invariant Schrödinger equations on the plane, Preprint. Available online at http://arxiv.org/abs/1405.1784 (2014)
Fanelli L., García A.: Counterexamples to Strichartz estimates for the magnetic Schrödinger equation. Commun. Contemp. Math. 13(2), 213–234 (2011)
Felli V., Ferrero A., Terracini S.: Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential. J. Eur. Math. Soc. 13(1), 119–174 (2011)
Felli V., Marchini E.M., Terracini S.: On Schrödinger operators with multipolar inverse-square potentials. J. Funct. Anal. 250, 265–316 (2007)
Felli V., Marchini E.M., Terracini S.: On Schrödinger operators with multisingular inverse-square anisotropic potentials. Indiana Univ. Math. J. 58, 617–676 (2009)
Georgiev V., Stefanov A., Tarulli M.: Smoothing—Strichartz estimates for the Schrödinger equation with small magnetic potential. Discret. Contin. Dyn. Syst. A 17, 771–786 (2007)
Goldberg M.: Dispersive estimates for the three-dimensional Schrödinger equation with rough potential. Am. J. Math. 128, 731–750 (2006)
Goldberg M., Schlag W.: Dispersive estimates for Schrödinger operators in dimensions one and three. Commun. Math. Phys. 251(1), 157–178 (2004)
Goldberg, M., Vega, L., Visciglia, N.: Counterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials. Int. Math. Res. Not. 2006 article ID 13927 (2006)
García Azorero J., Peral I.: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Equ. 144, 441–476 (1998)
Grillo G., Kovarik H.: Weighted dispersive estimates for two-dimensional Schrödinger operators with Aharonov-Bohm magnetic field. J. Differ. Equ. 256, 3889–3911 (2014)
Gurarie D.: Zonal Schrödinger operators on the n-Sphere: inverse spectral problem and rigidity. Commun. Math. Phys. 131, 571–603 (1990)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Reprint of the 1952 edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988)
Kalf, H., Schmincke, U.-W., Walter, J., Wüst, R.: On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials. Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens). Lecture Notes in Math., vol. 448, pp. 182–226. Springer, Berlin (1975)
Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin Heidelberg (1995)
Lebedev, N.N.: Special functions and their applications. Revised edition, translated from the Russian and edited by Richard A. Silverman. Unabridged and corrected republication. Dover Publications, Inc., New York (1972)
Landau L.J.: Bessel functions: monotonicity and bounds. J. Lond. Math. Soc. 61(1), 197–215 (2000)
Marzuola J., Metcalfe J., Tataru D.: Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations. J. Funct. Anal. 255, 1497–1553 (2008)
Planchon F., Stalker J., Tahvildar-Zadeh S.: Dispersive estimates for the wave equation with the inverse-square potential. Discret. Contin. Dyn. Syst. 9, 1387–1400 (2003)
Reed M., Simon B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press, New York (1975)
Robbiano, L., Zuily, C.: Strichartz estimates for Schrödinger equations with variable coefficients, Mém. Soc. Math. Fr. (N.S.) 101-102, vi+208 (2005)
Rodnianski I., Schlag W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155(3), 451–513 (2004)
Schlag W.: Dispersive estimates for Schrödinger operators: a survey, Mathematical aspects of nonlinear dispersive equations, 255285, Ann. of Math. Stud., 163. Princeton Univ. Press, Princeton (2007)
Simon B.: Essential self-adjointness of Schrödinger operators with singular potentials. Arch. Ration. Mech. Anal. 52, 44–48 (1973)
Staffilani G., Tataru D.: Strichartz estimates for a Schrödinger operator with nonsmooth coefficients. Commun. Partial Differ. Equ. 27(7-8), 1337–1372 (2002)
Stefanov A.: Strichartz estimates for the magnetic Schrödinger equation. Adv. Math. 210, 246–303 (2007)
Thomas L.E., Villegas-Blas C.: Singular continuous limiting eigenvalue distributions for Schrödinger operators on a 2-sphere. J. Funct. Anal. 141, 249–273 (1996)
Thomas L.E., Wassell S.R.: Semiclassical approximation for Schrödinger operators on a two-sphere at high energy. J. Math. Phys. 36(10), 5480–5505 (1995)
Weder R.: The W k,p -continuity of the Schrödinger wave operators on the line. Commun. Math. Phys. 208, 507–520 (1999)
Weder R.: \({L^p-L^{p^{\prime}}}\) estimates for the Schrödinger equations on the line and inverse scattering for the nonlinear Schrödinger equation with a potential. J. Funct. Anal. 170, 37–68 (2000)
Weinstein A.: Asymptotics for eigenvalue clusters for the laplacian plus a potencial. Duke Math. J. 44(4), 883–892 (1977)
Yajima K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110, 415–426 (1987)
Yajima K.: The W k,p-continuity of wave operators for Schrödinger operators. J. Math. Soc. Jpn. 47(3), 551–581 (1995)
Yajima K.: The W k,p-continuity of wave operators for Schrödinger operators III, even dimensional cases \({m\geqslant4}\). J. Math. Sci. Univ. Tokyo 2, 311–346 (1995)
Yajima K.: \({L\sp p}\) -boundedness of wave operators for two-dimensional Schrödinger operators. Commun. Math. Phys. 208(1), 125–152 (1999)
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Communicated by W. Schlag
L. Fanelli was supported by the Italian project FIRB 2012: “Dispersive dynamics: Fourier Analysis and Variational Methods”. V. Felli was partially supported by the P.R.I.N. 2012 grant “Variational and perturbative aspects of nonlinear differential problems”. M. A. Fontelos was supported by the Spanish project “MTM2011-26016”. A. Primo was supported by the Spanish project “MTM2010-18128”.
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Fanelli, L., Felli, V., Fontelos, M.A. et al. Time Decay of Scaling Invariant Electromagnetic Schrödinger Equations on the Plane. Commun. Math. Phys. 337, 1515–1533 (2015). https://doi.org/10.1007/s00220-015-2291-2
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DOI: https://doi.org/10.1007/s00220-015-2291-2