Sharp Hardy uncertainty principle and gaussian profiles of covariant Schrödinger evolutions

Cassano, B.; Fanelli, L.

doi:10.1090/S0002-9947-2014-06383-6

Sharp Hardy uncertainty principle and gaussian profiles of covariant Schrödinger evolutions
HTML articles powered by AMS MathViewer

by B. Cassano and L. Fanelli PDF

Trans. Amer. Math. Soc. 367 (2015), 2213-2233 Request permission

Abstract:

We prove a sharp version of the Hardy uncertainty principle for Schrödinger equations with external bounded electromagnetic potentials, based on logarithmic convexity properties of Schrödinger evolutions. We provide, in addition, an example of a real electromagnetic potential which produces the existence of solutions with critical gaussian decay, at two distinct times.

References

J. A. Barceló, L. Fanelli, S. Gutiérrez, A. Ruiz, and M. C. Vilela, Hardy uncertainty principle and unique continuation properties of covariant Schrödinger flows, J. Funct. Anal. 264 (2013), no. 10, 2386–2415. MR 3035060, DOI 10.1016/j.jfa.2013.02.017
Aline Bonami and Bruno Demange, A survey on uncertainty principles related to quadratic forms, Collect. Math. Vol. Extra (2006), 1–36. MR 2264204
M. Cowling, L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, The Hardy uncertainty principle revisited, Indiana Univ. Math. J. 59 (2010), no. 6, 2007–2025. MR 2919746, DOI 10.1512/iumj.2010.59.4395
J. Cruz-Sampedro, Unique continuation at infinity of solutions to Schrödinger equations with complex-valued potentials, Proc. Edinburgh Math. Soc. (2) 42 (1999), no. 1, 143–153. MR 1669361, DOI 10.1017/S0013091500020071
Hongjie Dong and Wolfgang Staubach, Unique continuation for the Schrödinger equation with gradient vector potentials, Proc. Amer. Math. Soc. 135 (2007), no. 7, 2141–2149. MR 2299492, DOI 10.1090/S0002-9939-07-08813-2
L. Erdős, Gaussian decay of the magnetic eigenfunctions, Geom. Funct. Anal. 6 (1996), no. 2, 231–248. MR 1384611, DOI 10.1007/BF02247886
L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, On uniqueness properties of solutions of Schrödinger equations, Comm. Partial Differential Equations 31 (2006), no. 10-12, 1811–1823. MR 2273975, DOI 10.1080/03605300500530446
L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, Convexity properties of solutions to the free Schrödinger equation with Gaussian decay, Math. Res. Lett. 15 (2008), no. 5, 957–971. MR 2443994, DOI 10.4310/MRL.2008.v15.n5.a10
L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, Hardy’s uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 883–907. MR 2443923, DOI 10.4171/JEMS/134
Luis Escauriaza, Carlos E. Kenig, Gustavo Ponce, and Luis Vega, The sharp Hardy uncertainty principle for Schrödinger evolutions, Duke Math. J. 155 (2010), no. 1, 163–187. MR 2730375, DOI 10.1215/00127094-2010-053
L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, Uncertainty principle of Morgan type and Schrödinger evolutions, J. Lond. Math. Soc. (2) 83 (2011), no. 1, 187–207. MR 2763951, DOI 10.1112/jlms/jdq072
Luca Fanelli and Luis Vega, Magnetic virial identities, weak dispersion and Strichartz inequalities, Math. Ann. 344 (2009), no. 2, 249–278. MR 2495769, DOI 10.1007/s00208-008-0303-7
Gerald B. Folland and Alladi Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (1997), no. 3, 207–238. MR 1448337, DOI 10.1007/BF02649110
Lars Hörmander, Linear partial differential operators, Springer-Verlag, Berlin-New York, 1976. MR 0404822
Alexandru D. Ionescu and Carlos E. Kenig, Uniqueness properties of solutions of Schrödinger equations, J. Funct. Anal. 232 (2006), no. 1, 90–136. MR 2200168, DOI 10.1016/j.jfa.2005.06.005
Akira Iwatsuka, Spectral representation for Schrödinger operators with magnetic vector potentials, J. Math. Kyoto Univ. 22 (1982/83), no. 2, 223–242. MR 666972, DOI 10.1215/kjm/1250521811
V. Z. Meshkov, On the possible rate of decrease at infinity of the solutions of second-order partial differential equations, Mat. Sb. 182 (1991), no. 3, 364–383 (Russian); English transl., Math. USSR-Sb. 72 (1992), no. 2, 343–361. MR 1110071, DOI 10.1070/SM1992v072n02ABEH001414
Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
A. Sitaram, M. Sundari, and S. Thangavelu, Uncertainty principles on certain Lie groups, Proc. Indian Acad. Sci. Math. Sci. 105 (1995), no. 2, 135–151. MR 1350473, DOI 10.1007/BF02880360
Elias M. Stein and Rami Shakarchi, Complex analysis, Princeton Lectures in Analysis, vol. 2, Princeton University Press, Princeton, NJ, 2003. MR 1976398