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:
[1]

P. AlbinC. GuillarmouL. 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.

[2]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33.

[3]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33.

[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.

[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.

[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.

[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.

[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.

[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.

[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)].

[11]

O. ImanuvilovG. 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.

[12]

O. Y. ImanuvilovG. 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.

[13]

J. Jost, Compact Riemann Surfaces, Universitext. Springer-Verlag, Berlin, third edition, 2006. An introduction to contemporary mathematics.

[14]

S. G. Krantz, Function Theory of Several Complex Variables, Pure and applied mathematics. Wiley, 1982.

[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.

[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.

[17]

M. Santacesaria, A Hölder-logarithmic stability estimate for an inverse problem in two dimensions, J. Inverse Ill-Posed Probl., 23 (2015), 51-73.

[18]

G. Schwarz, Hodge Decomposition-a Method for Solving Boundary Value Problems, volume 1607 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1995.

[19]

I. N. Vekua, Generalized Analytic Functions, International Series of Monographs in Pure and Applied Mathematics, Pergamon Press, 1962.

show all references

References:
[1]

P. AlbinC. GuillarmouL. 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.

[2]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33.

[3]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33.

[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.

[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.

[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.

[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.

[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.

[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.

[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)].

[11]

O. ImanuvilovG. 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.

[12]

O. Y. ImanuvilovG. 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.

[13]

J. Jost, Compact Riemann Surfaces, Universitext. Springer-Verlag, Berlin, third edition, 2006. An introduction to contemporary mathematics.

[14]

S. G. Krantz, Function Theory of Several Complex Variables, Pure and applied mathematics. Wiley, 1982.

[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.

[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.

[17]

M. Santacesaria, A Hölder-logarithmic stability estimate for an inverse problem in two dimensions, J. Inverse Ill-Posed Probl., 23 (2015), 51-73.

[18]

G. Schwarz, Hodge Decomposition-a Method for Solving Boundary Value Problems, volume 1607 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1995.

[19]

I. N. Vekua, Generalized Analytic Functions, International Series of Monographs in Pure and Applied Mathematics, Pergamon Press, 1962.

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