# American Institute of Mathematical Sciences

January  2018, 12(1): 1-28. doi: 10.3934/ipi.2018001

## Stability for a magnetic Schrödinger operator on a Riemann surface with boundary

 School of Mathematics and Statistics, University of Sydney, Sydney, Australia, 2006

* Corresponding author: leo.tzou@gmail.com

Received  March 2016 Revised  August 2017 Published  December 2017

Fund Project: The second author is supported by ARC Future Fellowship FT-130101346, the first author was employed by Vetenskapsrådet Project VR 170630

We consider a magnetic Schrödinger operator $(\nabla^X)^*\nabla^X+q$ on a compact Riemann surface with boundary and prove a $\log\log$-type stability estimate in terms of Cauchy data for the electric potential and magnetic field under the assumption that they satisfy appropriate a priori bounds. We also give a similar stability result for the holonomy of the connection 1-form $X$.

Citation: Joel Andersson, Leo Tzou. Stability for a magnetic Schrödinger operator on a Riemann surface with boundary. Inverse Problems & Imaging, 2018, 12 (1) : 1-28. doi: 10.3934/ipi.2018001
##### References:
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##### References:
 [1] P. Albin, C. Guillarmou, L. Tzou and G. Uhlmann, Inverse boundary problems for systems in two dimensions, Ann. Henri Poincaré, 14 (2013), 1551-1571.  doi: 10.1007/s00023-012-0229-1.  Google Scholar [2] A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33.   Google Scholar [3] A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33.  Google Scholar [4] O. Forster, Lectures on Riemann Surfaces, volume 81 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1991. Translated from the 1977 German original by Bruce Gilligan, Reprint of the 1981 English translation.  Google Scholar [5] C. Guillarmou and L. Tzou, Calderón inverse problem for the Schrödinger operator on Riemann surfaces, In The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis. Proceedings of the Workshop, Canberra, Australia, July 13-17,2009. , Canberra: Australian National University, Centre for Mathematics and its Applications, 44 (2010), 129-141.  Google Scholar [6] C. Guillarmou and L. Tzou, Calderón inverse problem with partial data on Riemann surfaces, Duke Math. J., 158 (2011), 83-120.  doi: 10.1215/00127094-1276310.  Google Scholar [7] C. Guillarmou and L. Tzou, Identification of a connection from Cauchy data on a Riemann surface with boundary, Geom. Funct. Anal., 21 (2011), 393-418.  doi: 10.1007/s00039-011-0110-2.  Google Scholar [8] C. Guillarmou and L. Tzou, The Calderón inverse problem in two dimensions, In Inverse Problems and Applications: Inside Out. II, volume 60 of Math. Sci. Res. Inst. Publ., pages 119-166. Cambridge Univ. Press, Cambridge, 2013.  Google Scholar [9] G. M. Henkin and R. G. Novikov, On the reconstruction of conductivity of a bordered two-dimensional surface in $\mathbb{R}^3$ from electrical current measurements on its boundary, J. Geom. Anal., 21 (2011), 543-587.  doi: 10.1007/s12220-010-9158-8.  Google Scholar [10] L. Hörmander, The Analysis of Linear Partial Differential Operators, I, Classics in Mathematics. Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m: 35001a)].  Google Scholar [11] O. Imanuvilov, G. Uhlmann and M. Yamamoto, Partial Cauchy data for general second order elliptic operators in two dimensions, Publ. Res. Inst. Math. Sci., 48 (2012), 971-1055.  doi: 10.2977/PRIMS/94.  Google Scholar [12] O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions, J. Amer. Math. Soc., 23 (2010), 655-691.  doi: 10.1090/S0894-0347-10-00656-9.  Google Scholar [13] J. Jost, Compact Riemann Surfaces, Universitext. Springer-Verlag, Berlin, third edition, 2006. An introduction to contemporary mathematics.  Google Scholar [14] S. G. Krantz, Function Theory of Several Complex Variables, Pure and applied mathematics. Wiley, 1982.  Google Scholar [15] R. G. Novikov and G. M. Khenkin, The $\overline\partial$-equation in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk, 42 (1987), 93-152,255.   Google Scholar [16] M. Santacesaria, New global stability estimates for the Calderón problem in two dimensions, J. Inst. Math. Jussieu, 12 (2013), 553-569.  doi: 10.1017/S147474801200076X.  Google Scholar [17] M. Santacesaria, A Hölder-logarithmic stability estimate for an inverse problem in two dimensions, J. Inverse Ill-Posed Probl., 23 (2015), 51-73.   Google Scholar [18] G. Schwarz, Hodge Decomposition-a Method for Solving Boundary Value Problems, volume 1607 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar [19] I. N. Vekua, Generalized Analytic Functions, International Series of Monographs in Pure and Applied Mathematics, Pergamon Press, 1962.  Google Scholar
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