Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Unique continuation for the Schrödinger equation with gradient term


Authors: Youngwoo Koh and Ihyeok Seo
Journal: Proc. Amer. Math. Soc. 146 (2018), 2555-2562
MSC (2010): Primary 35B60, 35B45; Secondary 35Q40
DOI: https://doi.org/10.1090/proc/13942
Published electronically: February 1, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain a unique continuation result for the differential inequality $ \vert (i\partial _t +\Delta )u \vert \leq \vert Vu\vert + \vert W\cdot \nabla u \vert$ by establishing $ L^2$ Carleman estimates. Here, $ V$ is a scalar function and $ W$ is a vector function, which may be time-dependent or time-independent. As a consequence, we give a similar result for the magnetic Schrödinger equation.


References [Enhancements On Off] (What's this?)

  • [1] 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, https://doi.org/10.1016/j.jfa.2013.02.017
  • [2] T. Carleman, Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat., Astr. Fys. 26 (1939), no. 17, 9 (French). MR 0000334
  • [3] Sagun Chanillo and Eric Sawyer, Unique continuation for $ \Delta+v$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc. 318 (1990), no. 1, 275-300. MR 958886, https://doi.org/10.2307/2001239
  • [4] Filippo Chiarenza and Alberto Ruiz, Uniform $ L^2$-weighted Sobolev inequalities, Proc. Amer. Math. Soc. 112 (1991), no. 1, 53-64. MR 1055768, https://doi.org/10.2307/2048479
  • [5] 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, https://doi.org/10.1090/S0002-9939-07-08813-2
  • [6] 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, https://doi.org/10.1080/03605300500530446
  • [7] L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, Uniqueness properties of solutions to Schrödinger equations, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 3, 415-442. MR 2917065, https://doi.org/10.1090/S0273-0979-2011-01368-4
  • [8] Alexandru D. Ionescu and Carlos E. Kenig, $ L^p$ Carleman inequalities and uniqueness of solutions of nonlinear Schrödinger equations, Acta Math. 193 (2004), no. 2, 193-239. MR 2134866, https://doi.org/10.1007/BF02392564
  • [9] 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, https://doi.org/10.1016/j.jfa.2005.06.005
  • [10] Victor Isakov, Carleman type estimates in an anisotropic case and applications, J. Differential Equations 105 (1993), no. 2, 217-238. MR 1240395, https://doi.org/10.1006/jdeq.1993.1088
  • [11] David Jerison and Carlos E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, with an appendix by E. M. Stein, Ann. of Math. (2) 121 (1985), no. 3, 463-494. MR 794370, https://doi.org/10.2307/1971205
  • [12] Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On unique continuation for nonlinear Schrödinger equations, Comm. Pure Appl. Math. 56 (2003), no. 9, 1247-1262. MR 1980854, https://doi.org/10.1002/cpa.10094
  • [13] C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 (1987), no. 2, 329-347. MR 894584, https://doi.org/10.1215/S0012-7094-87-05518-9
  • [14] Carlos E. Kenig and Christopher D. Sogge, A note on unique continuation for Schrödinger's operator, Proc. Amer. Math. Soc. 103 (1988), no. 2, 543-546. MR 943081, https://doi.org/10.2307/2047176
  • [15] Sanghyuk Lee and Ihyeok Seo, A note on unique continuation for the Schrödinger equation, J. Math. Anal. Appl. 389 (2012), no. 1, 461-468. MR 2876512, https://doi.org/10.1016/j.jmaa.2011.11.067
  • [16] Guozhen Lu and Thomas Wolff, Unique continuation with weak type lower order terms, Potential Anal. 7 (1997), no. 2, 603-614. MR 1467208, https://doi.org/10.1023/A:1017989619339
  • [17] Ihyeok Seo, Unique continuation for the Schrödinger equation with potentials in Wiener amalgam spaces, Indiana Univ. Math. J. 60 (2011), no. 4, 1203-1227. MR 2975341, https://doi.org/10.1512/iumj.2011.60.4824
  • [18] Ihyeok Seo, Carleman estimates for the Schrödinger operator and applications to unique continuation, Commun. Pure Appl. Anal. 11 (2012), no. 3, 1013-1036. MR 2968606, https://doi.org/10.3934/cpaa.2012.11.1013
  • [19] Ihyeok Seo, Global unique continuation from a half space for the Schrödinger equation, J. Funct. Anal. 266 (2014), no. 1, 85-98. MR 3121722, https://doi.org/10.1016/j.jfa.2013.09.025
  • [20] Ihyeok Seo, Unique continuation for fractional Schrödinger operators in three and higher dimensions, Proc. Amer. Math. Soc. 143 (2015), no. 4, 1661-1664. MR 3314078, https://doi.org/10.1090/S0002-9939-2014-12594-9
  • [21] Ihyeok Seo, Carleman inequalities for fractional Laplacians and unique continuation, Taiwanese J. Math. 19 (2015), no. 5, 1533-1540. MR 3412019, https://doi.org/10.11650/tjm.19.2015.5624
  • [22] Ihyeok Seo, From resolvent estimates to unique continuation for the Schrödinger equation, Trans. Amer. Math. Soc. 368 (2016), no. 12, 8755-8784. MR 3551588, https://doi.org/10.1090/tran/6635
  • [23] Thomas H. Wolff, Unique continuation for $ \vert\Delta u\vert\le V\vert\nabla u\vert$ and related problems, Rev. Mat. Iberoamericana 6 (1990), no. 3-4, 155-200. MR 1125760, https://doi.org/10.4171/RMI/101
  • [24] T. H. Wolff, A property of measures in $ {\bf R}^N$ and an application to unique continuation, Geom. Funct. Anal. 2 (1992), no. 2, 225-284. MR 1159832, https://doi.org/10.1007/BF01896975
  • [25] Bing-Yu Zhang, Unique continuation properties of the nonlinear Schrödinger equation, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 1, 191-205. MR 1433092, https://doi.org/10.1017/S0308210500023581

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35B60, 35B45, 35Q40

Retrieve articles in all journals with MSC (2010): 35B60, 35B45, 35Q40


Additional Information

Youngwoo Koh
Affiliation: Department of Mathematics Education, Kongju National University, Kongju 32588, Republic of Korea
Email: ywkoh@kongju.ac.kr

Ihyeok Seo
Affiliation: Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
Email: ihseo@skku.edu

DOI: https://doi.org/10.1090/proc/13942
Keywords: Unique continuation, Carleman estimates, Schr\"odinger equation.
Received by editor(s): June 9, 2017
Received by editor(s) in revised form: August 28, 2017, and September 1, 2017
Published electronically: February 1, 2018
Additional Notes: The first author was supported by NRF Grant 2016R1D1A1B03932049 (Republic of Korea). The second author was supported by the NRF grant funded by the Korea government(MSIP) (No. 2017R1C1B5017496).
Communicated by: Joachim Krieger
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society