Article Contents
Article Contents

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

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 $X$.

Mathematics Subject Classification: Primary: 35, 51; Secondary: 58.

 Citation:

•  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. A. L. Bukhgeim , Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008) , 19-33. A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33. 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. 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. 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. 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. 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. 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. 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)]. 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. 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. J. Jost, Compact Riemann Surfaces, Universitext. Springer-Verlag, Berlin, third edition, 2006. An introduction to contemporary mathematics. S. G. Krantz, Function Theory of Several Complex Variables, Pure and applied mathematics. Wiley, 1982. 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. 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. M. Santacesaria , A Hölder-logarithmic stability estimate for an inverse problem in two dimensions, J. Inverse Ill-Posed Probl., 23 (2015) , 51-73. G. Schwarz, Hodge Decomposition-a Method for Solving Boundary Value Problems, volume 1607 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1995. I. N. Vekua, Generalized Analytic Functions, International Series of Monographs in Pure and Applied Mathematics, Pergamon Press, 1962.