## An inverse boundary value problem for Schrödinger operators with vector potentials

HTML articles powered by AMS MathViewer

- by Zi Qi Sun PDF
- Trans. Amer. Math. Soc.
**338**(1993), 953-969 Request permission

## Abstract:

We consider the Schrödinger operator for a magnetic potential $\vec A$ and an electric potential $q$, which are supported in a bounded domain in ${\mathbb {R}^n}$ with $n \geq 3$. We prove that knowledge of the Dirichlet to Neumann map associated to the Schrödinger operator determines the magnetic field $\operatorname {rot}(\vec A)$ and the electric potential $q$ simultaneously, provided $\operatorname {rot}(\vec A)$ is small in the ${L^\infty }$ topology.## References

- Giovanni Alessandrini,
*Stable determination of conductivity by boundary measurements*, Appl. Anal.**27**(1988), no. 1-3, 153–172. MR**922775**, DOI 10.1080/00036818808839730 - Alberto-P. Calderón,
*On an inverse boundary value problem*, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) Soc. Brasil. Mat., Rio de Janeiro, 1980, pp. 65–73. MR**590275** - Sagun Chanillo,
*A problem in electrical prospection and an $n$-dimensional Borg-Levinson theorem*, Proc. Amer. Math. Soc.**108**(1990), no. 3, 761–767. MR**998731**, DOI 10.1090/S0002-9939-1990-0998731-1 - Victor Isakov,
*Completeness of products of solutions and some inverse problems for PDE*, J. Differential Equations**92**(1991), no. 2, 305–316. MR**1120907**, DOI 10.1016/0022-0396(91)90051-A - Robert Kohn and Michael Vogelius,
*Determining conductivity by boundary measurements*, Comm. Pure Appl. Math.**37**(1984), no. 3, 289–298. MR**739921**, DOI 10.1002/cpa.3160370302 - R. V. Kohn and M. Vogelius,
*Determining conductivity by boundary measurements. II. Interior results*, Comm. Pure Appl. Math.**38**(1985), no. 5, 643–667. MR**803253**, DOI 10.1002/cpa.3160380513
R. B. Lavine and A. Nachman, - Adrian I. Nachman,
*Reconstructions from boundary measurements*, Ann. of Math. (2)**128**(1988), no. 3, 531–576. MR**970610**, DOI 10.2307/1971435 - Adrian Nachman, John Sylvester, and Gunther Uhlmann,
*An $n$-dimensional Borg-Levinson theorem*, Comm. Math. Phys.**115**(1988), no. 4, 595–605. MR**933457**, DOI 10.1007/BF01224129 - R. G. Novikov and G. M. Khenkin,
*The $\overline \partial$-equation in the multidimensional inverse scattering problem*, Uspekhi Mat. Nauk**42**(1987), no. 3(255), 93–152, 255 (Russian). MR**896879** - A. G. Ramm,
*Recovery of the potential from fixed-energy scattering data*, Inverse Problems**4**(1988), no. 3, 877–886. MR**965652**, DOI 10.1088/0266-5611/4/3/020 - John Sylvester and Gunther Uhlmann,
*A global uniqueness theorem for an inverse boundary value problem*, Ann. of Math. (2)**125**(1987), no. 1, 153–169. MR**873380**, DOI 10.2307/1971291 - John Sylvester and Gunther Uhlmann,
*A uniqueness theorem for an inverse boundary value problem in electrical prospection*, Comm. Pure Appl. Math.**39**(1986), no. 1, 91–112. MR**820341**, DOI 10.1002/cpa.3160390106 - Zi Qi Sun,
*The inverse conductivity problem in two dimensions*, J. Differential Equations**87**(1990), no. 2, 227–255. MR**1072901**, DOI 10.1016/0022-0396(90)90002-7 - Zi Qi Sun,
*An inverse boundary value problem for the Schrödinger operator with vector potentials in two dimensions*, Comm. Partial Differential Equations**18**(1993), no. 1-2, 83–124. MR**1211726**, DOI 10.1080/03605309308820922 - Zi Qi Sun and Gunther Uhlmann,
*Generic uniqueness for an inverse boundary value problem*, Duke Math. J.**62**(1991), no. 1, 131–155. MR**1104326**, DOI 10.1215/S0012-7094-91-06206-X - Zi Qi Sun and Gunther Uhlmann,
*Inverse scattering for singular potentials in two dimensions*, Trans. Amer. Math. Soc.**338**(1993), no. 1, 363–374. MR**1126214**, DOI 10.1090/S0002-9947-1993-1126214-4 - Ricardo Weder,
*Global uniqueness at fixed energy in multidimensional inverse scattering theory*, Inverse Problems**7**(1991), no. 6, 927–938. MR**1140323**, DOI 10.1088/0266-5611/7/6/012

*Global uniqueness in inverse problems with singular potentials*, in preparation.

## Additional Information

- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**338**(1993), 953-969 - MSC: Primary 35J10; Secondary 35R30, 47N50, 81Q05, 81V10
- DOI: https://doi.org/10.1090/S0002-9947-1993-1179400-1
- MathSciNet review: 1179400