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2017 | OriginalPaper | Chapter

Spherical Schrödinger Hamiltonians: Spectral Analysis and Time Decay

Author : Luca Fanelli

Published in: Advances in Quantum Mechanics

Publisher: Springer International Publishing

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Abstract

In this survey, we review recent results concerning the canonical dispersive flow e ^itH led by a Schrödinger Hamiltonian H. We study, in particular, how the time decay of space L ^p-norms depends on the frequency localization of the initial datum with respect to the some suitable spherical expansion. A quite complete description of the phenomenon is given in terms of the eigenvalues and eigenfunctions of the restriction of H to the unit sphere, and a comparison with some uncertainty inequality is presented.

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previous chapter Logarithmic Sobolev Inequalities for an Ideal Bose Gas

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M. Beceanu, M. Goldberg, Schrödinger dispersive estimates for a scaling-critical class of potentials. Commun. Math. Phys. 314, 471–481 (2012)CrossRefMATH

A. Bezubik, A. Strasburger, A new form of the spherical expansion of zonal functions and Fourier transforms of \(\mathop{\mathrm{SO}}(d)\)-finite functions. SIGMA Symmetry Integrability Geom. Methods Appl. 2, Paper 033, 8 pp. (2006)

G. Borg, Umkerhrung der Sturm-Liouvillischen Eigebnvertanfgabe Bestimung der difúerentialgleichung die Eigenverte. Acta Math. 78, 1–96 (1946)MathSciNetCrossRef

N. Burq, F. Planchon, J. Stalker, S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential. J. Funct. Anal. 203(2), 519–549 (2003)MathSciNetCrossRefMATH

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

T. Cazenave, Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York (American Mathematical Society, Providence, 2003)

P. D’Ancona, L. Fanelli, L ^p-boundedness of the wave operator for the one dimensional Schrödinger operators. Commun. Math. Phys. 268, 415–438 (2006)CrossRefMATH

P. D’Ancona, L. Fanelli, Decay estimates for the wave and Dirac equations with a magnetic potential. Comm. Pure Appl. Math. 60, 357–392 (2007)MathSciNetCrossRefMATH

T. Duyckaerts, Inégalités de résolvante pour l’opérateur de Schrödinger avec potentiel multipolaire critique. Bulletin de la Société mathématique de France 134, 201–239 (2006)MathSciNetCrossRefMATH

10.

M.B. Erdogan, M. Goldberg, W. Schlag, Strichartz and smoothing estimates for Schrodinger operators with large magnetic potentials in \(\mathbb{R}^{3}\). J. Eur. Math. Soc. 10, 507–531 (2008)MathSciNetCrossRefMATH

11.

M.B. Erdogan, M. Goldberg, W. Schlag, Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions. Forum Math. 21, 687–722 (2009)MathSciNetCrossRefMATH

12.

L. Fanelli, V. Felli, M. Fontelos, A. Primo, Time decay of scaling critical electromagnetic Schrödinger flows. Commun. Math. Phys. 324, 1033–1067 (2013)CrossRefMATH

13.

L. Fanelli, V. Felli, M. Fontelos, A. Primo, Time decay of scaling invariant electromagnetic Schrödinger equations on the plane. Commun. Math. Phys. 337, 1515–1533 (2015)CrossRefMATH

14.

L. Fanelli, V. Felli, M. Fontelos, A. Primo, Frequency-dependent time decay of Schrödinger flows. J. Spectral Theory (to appear in)

15.

L. Fanelli, G. Grillo, H. Kovařík, Improved time-decay for a class of scaling critical electromagnetic Schrödinger flows. J. Funct. Anal. 269, 3336–3346 (2015)MathSciNetCrossRefMATH

16.

V. Felli, A. Ferrero, S. Terracini, Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential. J. Eur. Math. Soc. 13(1), 119–174 (2011)MathSciNetCrossRefMATH

17.

M. Goldberg, Dispersive estimates for the three-dimensional schrödinger equation with rough potential. Am. J. Math. 128, 731–750 (2006)CrossRefMATH

18.

M. Goldberg, W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three. Commun. Math. Phys. 251(1), 157–178 (2004)CrossRefMATH

19.

G. Grillo, H. Kovarik, Weighted dispersive estimates for two-dimensional Schrödinger operators with Aharonov-Bohm magnetic field. J. Differ. Equ. 256, 3889–3911 (2014)CrossRefMATH

20.

D. Gurarie, Zonal Schrödinger operators on the n-Sphere: inverse spectral problem and rRigidity. Commun. Math. Phys. 131 (1990), 571–603CrossRefMATH

21.

M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and Its Applications, vol. 98 (Cambridge University Press, Cambridge, 2005)

22.

H. Kalf, U.-W. Schmincke, J. Walter, R. Wüst, 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 (Springer, Berlin, 1975), pp. 182–226

23.

M. Keel, T. Tao, Endpoint strichartz estimates. Am. J. Math. 120(5), 955–980 (1998)MathSciNetCrossRefMATH

24.

A. Laptev, T. Weidl, Hardy inequalities for magnetic Dirichlet forms. Mathematical results in quantum mechanics (Prague, 1998), 299–305; Oper. Theory Adv. Appl. 108, (Birkhäuser, Basel, 1999)

25.

F. Planchon, J. Stalker, S. Tahvildar-Zadeh, Dispersive estimates for the wave equation with the inverse-square potential. Discrete Contin. Dyn. Syst. 9, 1387–1400 (2003)MathSciNetCrossRefMATH

26.

I. Rodnianski, W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155(3), 451–513 (2004)MathSciNetCrossRefMATH

27.

W. Schlag, Dispersive estimates for Schrödinger operators: a survey. Mathematical Aspects of Nonlinear Dispersive Equations, 255285, Ann. of Math. Stud., vol. 163 (Princeton University Press, Princeton, 2007)

28.

B. Simon, Essential self-adjointness of Schrödinger operators with singular potentials. Arch. Ration. Mech. Anal. 52, 44–48 (1973)CrossRefMATH

29.

L.E. Thomas, C. Villegas-Blas, Singular continuous limiting eigenvalue distributions for Schrödinger operators on a 2-sphere. J. Funct. Anal. 141, 249–273 (1996)MathSciNetCrossRefMATH

30.

L.E. Thomas, S.R. Wassell, Semiclassical Approximation for Schrödinger operators on a two-sphere at high energy. J. Math. Phys. 36(10), 5480–5505 (1995)MathSciNetCrossRefMATH

31.

R. Weder, The W _k, p-continuity of the Schrödinger wave operators on the line. Commun. Math. Phys. 208, 507–520 (1999)CrossRefMATH

32.

R. Weder, \(L^{p} - L^{p^{{\prime}} }\) 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)

33.

A. Weinstein, Asymptotics for eigenvalue clusters for the laplacian plus a potencial. Duke Math. J. 44(4), 883..892 (1977)

34.

K. Yajima, Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110, 415–426 (1987)CrossRefMATH

35.

K. Yajima, The W ^k, p-continuity of wave operators for Schrödinger operators. J. Math. Soc. Jpn. 47(3), 551–581 (1995)CrossRefMATH

36.

K. Yajima, The W ^k, p-continuity of wave operators for Schrödinger operators III, even dimensional cases \(m\geqslant 4\). J. Math. Sci. Univ. Tokyo 2, 311–346 (1995)MathSciNetMATH

37.

K. Yajima, L ^p-boundedness of wave operators for two-dimensional Schrödinger operators. Commun. Math. Phys. 208(1), 125–152 (1999)MathSciNetCrossRefMATH

Title: Spherical Schrödinger Hamiltonians: Spectral Analysis and Time Decay
Author: Luca Fanelli
Publisher: Springer International Publishing
Book: Advances in Quantum Mechanics
Print ISBN: 978-3-319-58903-9

Electronic ISBN: 978-3-319-58904-6

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-58904-6_8

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