Abstract
We review different properties related to the Cauchy problem for the (nonlinear) Schrödinger equation with a smooth potential. For energy-subcritical nonlinearities and at most quadratic potentials, we investigate the necessary decay in space in order for the Cauchy problem to be locally (and globally) well posed. The characterization of the minimal decay is different in the case of super-quadratic potentials.
References
Ben Abdallah N., Castella F., Méhats F.: Time averaging for the strongly confined nonlinear Schrödinger equation, using almost periodicity. J. Differ. Equ. 245, 154–200 (2008)
Bourgain J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3, 107–156 (1993)
Burq N., Gérard P., Tzvetkov N.: Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Am. J. Math. 126, 569–605 (2004)
Carles R.: WKB analysis for nonlinear Schrödinger equations with potential. Commun. Math. Phys. 269, 195–221 (2007)
Carles R.: On the Cauchy problem in Sobolev spaces for nonlinear Schrödinger equations with potential. Portugal. Math. (N. S.) 65, 191–209 (2008)
Carles R.: Nonlinear Schrödinger equation with time dependent potential. Commun. Math. Sci. 9, 937–964 (2011)
Cazenave, T.: Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York (2003)
Cazenave T., Weissler F.: The Cauchy problem for the critical nonlinear Schrödinger equation in H s. Nonlinear Anal. TMA 14, 807–836 (1990)
D’Ancona P., Fanelli L.: Smoothing estimates for the Schrödinger equation with unbounded potentials. J. Differ. Equ. 246, 4552–4567 (2009)
Dunford, N., Schwartz, J.T.: Linear operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space. With the assistance of William G. Bade and Robert G. Bartle. Interscience Publishers, Wiley, New York, London (1963)
Fujiwara D.: A construction of the fundamental solution for the Schrödinger equation. J. Analyse Math. 35, 41–96 (1979)
Ginibre J., Velo G.: On a class of nonlinear Schrödinger equations. I The Cauchy problem, general case. J. Funct. Anal. 32, 1–32 (1979)
Kitada H.: On a construction of the fundamental solution for Schrödinger equations. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 193–226 (1980)
Mizutani H.: Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials. Commun. Pure Appl. Anal. 13, 2177–2210 (2014)
Oh Y.-G.: Cauchy problem and Ehrenfest’s law of nonlinear Schrödinger equations with potentials. J. Differ. Equ. 81, 255–274 (1989)
Pitaevskii, L., Stringari, S.: Bose–Einstein Condensation, vol. 116 of International Series of Monographs on Physics. The Clarendon Press Oxford University Press, Oxford (2003)
Reed M., Simon B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1975)
Robbiano L., Zuily C.: Remark on the Kato smoothing effect for Schrödinger equation with superquadratic potentials. Commun. Partial Differ. Equ. 33, 718–727 (2008)
Sulem C., Sulem P.-L.: The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse. Springer, New York (1999)
Thomann L.: A remark on the Schrödinger smoothing effect. Asymptot. Anal. 69, 117–123 (2010)
Yajima K.: Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations. Commun. Math. Phys. 181, 605–629 (1996)
Yajima K., Zhang G.: Smoothing property for Schrödinger equations with potential superquadratic at infinity. Commun. Math. Phys. 221, 573–590 (2001)
Yajima K., Zhang G.: Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity. J. Differ. Equ. 202, 81–110 (2004)
Zelditch S.: Reconstruction of singularities for solutions of Schrödinger’s equation. Commun. Math. Phys. 90, 1–26 (1983)
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This work was supported by the French ANR projects SchEq (ANR-12-JS01-0005-01) and BECASIM (ANR-12-MONU-0007-04).
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Carles, R. Sharp weights in the Cauchy problem for nonlinear Schrödinger equations with potential. Z. Angew. Math. Phys. 66, 2087–2094 (2015). https://doi.org/10.1007/s00033-015-0501-6
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DOI: https://doi.org/10.1007/s00033-015-0501-6