A unique continuation theorem for the Schrödinger equation with singular magnetic field

Kurata, Kazuhiro

doi:10.1090/S0002-9939-97-03672-1

A unique continuation theorem for the Schrödinger equation with singular magnetic field

Author: Kazuhiro Kurata
Journal: Proc. Amer. Math. Soc. 125 (1997), 853-860
MSC (1991): Primary 35B60, 35J10, 35Q60
DOI: https://doi.org/10.1090/S0002-9939-97-03672-1
MathSciNet review: 1363173
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show a unique continuation theorem for the Schrödinger equation $(\frac {1}{i}\nabla -\mathbf {A})^2 u+ Vu=0$ with singular coefficients $\mathbf {A}$ and $V$.

B. Barceló, C. E. Kenig, A. Ruiz, and C. D. Sogge, Weighted Sobolev inequalities and unique continuation for the Laplacian plus lower order terms, Illinois J. Math. 32 (1988), no. 2, 230–245. MR 945861
Michael S. P. Eastham and Hubert Kalf, Schrödinger-type operators with continuous spectra, Research Notes in Mathematics, vol. 65, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667015
Eugene B. Fabes, Nicola Garofalo, and Fang-Hua Lin, A partial answer to a conjecture of B. Simon concerning unique continuation, J. Funct. Anal. 88 (1990), no. 1, 194–210. MR 1033920, DOI https://doi.org/10.1016/0022-1236%2890%2990125-5
Nicola Garofalo and Fang-Hua Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J. 35 (1986), no. 2, 245–268. MR 833393, DOI https://doi.org/10.1512/iumj.1986.35.35015
Nicola Garofalo and Fang-Hua Lin, Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure Appl. Math. 40 (1987), no. 3, 347–366. MR 882069, DOI https://doi.org/10.1002/cpa.3160400305
Hubert Kalf, Une remarque au sujet de prolongement unique des solutions de l’équation de Schrödinger, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 10, 579–581 (French, with English summary). MR 685029
Lars Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations 8 (1983), no. 1, 21–64. MR 686819, DOI https://doi.org/10.1080/03605308308820262
Kazuhiro Kurata, A unique continuation theorem for uniformly elliptic equations with strongly singular potentials, Comm. Partial Differential Equations 18 (1993), no. 7-8, 1161–1189. MR 1233189, DOI https://doi.org/10.1080/03605309308820968

Kurata, K. Local boundedness and continuity for weak solutions of $-(\nabla - i{ \mathbf {b}})^2 u+ Vu=0$. Math. Z. (to appear).

Christopher D. Sogge, Strong uniqueness theorems for second order elliptic differential equations, Amer. J. Math. 112 (1990), no. 6, 943–984. MR 1081811, DOI https://doi.org/10.2307/2374732

Thomas H. Wolff, Unique continuation for $|\Delta u|\le V|\nabla u|$ and related problems, Rev. Mat. Iberoamericana 6 (1990), no. 3-4, 155–200. MR 1125760, DOI https://doi.org/10.4171/RMI/101

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, DOI https://doi.org/10.1007/BF01896975