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Unique continuation for fractional Schrödinger operators in three and higher dimensions


Author: Ihyeok Seo
Journal: Proc. Amer. Math. Soc. 143 (2015), 1661-1664
MSC (2010): Primary 35B60; Secondary 35J10
DOI: https://doi.org/10.1090/S0002-9939-2014-12594-9
Published electronically: December 1, 2014
MathSciNet review: 3314078
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Abstract: We prove the unique continuation property for the differential inequality $ \vert(-\Delta )^{\alpha /2}u\vert\leq \vert V(x)u\vert$, where $ 0<\alpha <n$ and $ V\in L_{\textrm {loc}}^{n/\alpha ,\infty }(\mathbb{R}^n)$, $ n\geq 3$.


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  • [1] David Jerison and Carlos E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2) 121 (1985), no. 3, 463-494. With an appendix by E. M. Stein. MR 794370 (87a:35058), https://doi.org/10.2307/1971205
  • [2] Carlos E. Kenig and Nikolai Nadirashvili, A counterexample in unique continuation, Math. Res. Lett. 7 (2000), no. 5-6, 625-630. MR 1809288 (2001m:35044), https://doi.org/10.4310/MRL.2000.v7.n5.a8
  • [3] Herbert Koch and Daniel Tataru, Sharp counterexamples in unique continuation for second order elliptic equations, J. Reine Angew. Math. 542 (2002), 133-146. MR 1880829 (2002m:35020), https://doi.org/10.1515/crll.2002.003
  • [4] Izabella Łaba, Unique continuation for Schrödinger operators and for higher powers of the Laplacian, Math. Methods Appl. Sci. 10 (1988), no. 5, 531-542. MR 965420 (89k:35063), https://doi.org/10.1002/mma.1670100504
  • [5] Nikolai Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), no. 4-6, 298-305. MR 1755089 (2000m:81097), https://doi.org/10.1016/S0375-9601(00)00201-2
  • [6] Ihyeok Seo, Remark on unique continuation for higher powers of the Laplace operator, J. Math. Anal. Appl. 397 (2013), no. 2, 766-771. MR 2979611, https://doi.org/10.1016/j.jmaa.2012.08.035
  • [7] Ihyeok Seo, On unique continuation for Schrödinger operators of fractional and higher orders, Math. Nachr. 287 (2014), no. 5-6, 699-703. MR 3193945, https://doi.org/10.1002/mana.201300008
  • [8] Elias M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482-492. MR 0082586 (18,575d)
  • [9] E. M. Stein, Appendix to ``unique continuation'', Ann. of Math. 121 (1985), 489-494.

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Additional Information

Ihyeok Seo
Affiliation: Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea
Email: ihseo@skku.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12594-9
Keywords: Unique continuation, Schr\"odinger operators
Received by editor(s): September 5, 2013
Published electronically: December 1, 2014
Communicated by: Joachim Krieger
Article copyright: © Copyright 2014 American Mathematical Society

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