Unique determination of the electric potential in the presence of a fixed magnetic potential in the plane

Caro, Pedro; Rogers, Keith M.

Journées équations aux dérivées partielles (2018), Exposé no. 7, 9 p.

For potentials $V\in L^{\infty }({ℝ}^2,ℝ)$ and $A\in W^{1,\infty }({ℝ}^2,{ℝ}^2)$ with compact support, we consider the Schrödinger equation $-{(\nabla +iA)}^{2}u+Vu=k^{2}u$ with fixed positive energy $k^2$. Under a mild additional regularity hypothesis, and with fixed magnetic potential $A$, we show that the scattering solutions uniquely determine the electric potential $V$. For this we develop the method of Bukhgeim for the purely electric Schrödinger equation.

Classification : 35P25, 45Q05, 35J10

[1] Agaltsov, Alexey D. A global uniqueness result for acoustic tomography of moving fluid, Bull. Sci. Math., Volume 139 (2015) no. 8, pp. 937-942 | Zbl

[2] Albin, Pierre; Guillarmou, Colin; Tzou, Leo; Uhlmann, Gunther Inverse boundary problems for systems in two dimensions, Ann. Henri Poincaré, Volume 14 (2013) no. 6, pp. 1551-1571 | Zbl

[3] Astala, Kari; Faraco, Daniel; Rogers, Keith M. Recovery of the Dirichlet-to-Neumann map from scattering data in the plane, RIMS Kôkyûroku Bessatsu, Volume 49 (2014), pp. 65-73 | Zbl

[4] Astala, Kari; Faraco, Daniel; Rogers, Keith M. Unbounded potential recovery in the plane, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016) no. 5, pp. 1027-1051 | Zbl

[5] Bergh, Jöran; Löfström, Jörgen Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer, 1976 | Zbl

[6] Blåsten, Eemeli On the Gel’fand–Calderón inverse problem in two dimensions, University of Helsinki (Finland) (2013) (Ph. D. Thesis)

[7] Blåsten, Eemeli; Imanuvilov, Oleg; Yamamoto, Masahiro Stability and Uniqueness for a two-dimensional inverse boundary value problem for less regular potentials, Inverse Probl. Imaging, Volume 9 (2015) no. 3, pp. 709-723 | Zbl

[8] Blåsten, Eemeli; Tzou, Leo; Wang, Jenn-Nan Uniqueness for the inverse boundary value problem with singular potentials in 2D (2018) (https://arxiv.org/abs/1704.06397)

[9] Bukhgeǐm, Alexandre L. Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., Volume 16 (2008) no. 1, pp. 19-33 | Zbl

[10] Eskin, Gregory; Ralston, James V. Inverse scattering problem for the Schrödinger equation with magnetic potential at fixed energy, Commun. Math. Phys., Volume 173 (1995), pp. 199-224 | Zbl

[11] Eskin, Gregory; Ralston, James V. Inverse scattering problems for Schrödinger operators with magnetic and electric potentials, Inverse problems in wave propagation (The IMA Volumes in Mathematics and its Applications), Volume 90, Springer, 1997, pp. 147-166 | Zbl

[12] Guillarmou, Colin; Tzou, Leo Identification of a connection from Cauchy data on a Riemann surface with boundary, Geom. Funct. Anal., Volume 21 (2011) no. 2, pp. 393-418 | Zbl

[13] Haberman, Boaz Unique determination of a magnetic Schrödinger operator with unbounded magnetic potential from boundary data, Int. Math. Res. Not., Volume 2018 (2018) no. 4, pp. 1080-1128 | Zbl

[14] Imanuvilov, Oleg; Uhlmann, Gunther; Yamamoto, Masahiro Partial Cauchy data for general second order elliptic operators in two dimensions, Publ. Res. Inst. Math. Sci., Volume 48 (2012) no. 4, pp. 971-1055 | Zbl

[15] Isakov, Victor; Nachman, Adrian Global uniqueness for a two-dimensional semilinear elliptic inverse problem, Trans. Am. Math. Soc., Volume 347 (1995) no. 9, pp. 3375-3390 | Zbl

[16] Krupchyk, Katsiaryna; Uhlmann, Gunther Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential, Commun. Math. Phys., Volume 327 (2014) no. 3, pp. 993-1009 | Zbl

[17] Lakshtanov, Evgeny; Tejero, Jorge; Vainberg, Boris Uniqueness in the inverse conductivity problem for complex-valued Lipschitz conductivities in the plane, SIAM J. Math. Anal., Volume 49 (2017) no. 5, pp. 3766-3775 | Zbl

[18] Lakshtanov, Evgeny; Vainberg, Boris Recovery of $L^p$-potential in the plane, J. Inverse Ill-Posed Probl., Volume 25 (2017), pp. 633-651 | Zbl

[19] Nachman, Adrian Global uniqueness for a two-dimensional inverse boundary value problem, Ann. Math., Volume 142 (1995) no. 1, pp. 71-96 | Zbl

[20] Nachman, Adrian; Sylvester, John; Uhlmann, Gunther An $n$-dimensional Borg-Levinson theorem, Commun. Math. Phys., Volume 115 (1988) no. 4, pp. 595-605 | Zbl

[21] Nakamura, Gen; Sun, Ziqi; Uhlmann, Gunther Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., Volume 303 (1995) no. 3, pp. 377-388 | Zbl

[22] Päivärinta, Lassi; Salo, Mikko; Uhlmann, Gunther Inverse scattering for the magnetic Schrödinger operator, J. Funct. Anal., Volume 259 (2010) no. 7, pp. 1771-1798 | Zbl

[23] Salo, Mikko Inverse problems for nonsmooth first order perturbations of the Laplacian, Annales Academiæ Scientiarum Fennicæ. Mathematica. Dissertationes, 139, Suomalainen Tiedeakatemia, 2004 | Zbl

[24] Salo, Mikko Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field, Commun. Partial Differ. Equations, Volume 31 (2006) no. 11, pp. 1639-1666 | Zbl

[25] Salo, Mikko Inverse boundary value problems for the magnetic Schrödinger equation, Proceedings Inverse Problems in Applied Sciences (Sapporo 2006) (Journal of Physics: Conference Series), Volume 73, IOP Publishing, 2007 | DOI

[26] dos Santos Ferreira, David; Kenig, Carlos E.; Sjöstrand, Johannes; Uhlmann, Gunther Determining a Magnetic Schrödinger Operator from Partial Cauchy Data, Commun. Math. Phys., Volume 271 (2007) no. 2, pp. 467-488 | Zbl

[27] Stein, Elias M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, 1993 | Zbl

[28] Sun, Ziqi An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Am. Math. Soc., Volume 338 (1993) no. 2, pp. 953-969 | Zbl

[29] Tejero, Jorge Reconstruction and stability for piecewise smooth potentials in the plane, SIAM J. Math. Anal., Volume 49 (2017) no. 1, pp. 398-420 | Zbl

[30] Tejero, Jorge Reconstruction of rough potentials in the plane (2018) (https://arxiv.org/abs/1811.09481)

[31] Tejero, Jorge On the method of Bukhgeim for two-dimensional inverse problems, Universidad Autónoma de Madrid (Spain) (2019) (Ph. D. Thesis)