We study inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective.
S.Bhattacharyya, An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data, Inverse Probl. Imaging12(3) (2018), 801–830. doi:10.3934/ipi.2018034.
2.
M.Cekić, The Calderón problem for connections, Commun. Partial. Differ. Equ.42(11) (2017), 1781–1836. doi:10.1080/03605302.2017.1390678.
3.
F.J.Chung, Partial data for the Neumann–Dirichlet magnetic Schrödinger inverse problem, Inverse Probl. Imaging8(4) (2014), 959–989. doi:10.3934/ipi.2014.8.959.
4.
F.J.Chung, A partial data result for the magnetic Schrödinger inverse problem, Anal. PDE7(1) (2014), 117–157. doi:10.2140/apde.2014.7.117.
5.
D.Dos Santos Ferreira, C.Kenig and M.Salo, Determining an unbounded potential from Cauchy data in admissible geometries, Commun. Partial. Differ. Equ.38(1) (2013), 50–68. doi:10.1080/03605302.2012.736911.
6.
D.Dos Santos Ferreira, C.Kenig, M.Salo and G.Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math.178(1) (2009), 119–171. doi:10.1007/s00222-009-0196-4.
7.
D.Dos Santos Ferreira, C.Kenig, J.Sjöstrand and G.Uhlmann, Determining a magnetic Schrödinger operator from partial Cauchy data, Commun. Math. Phys.271(2) (2007), 467–488. doi:10.1007/s00220-006-0151-9.
8.
D.Dos Santos Ferreira, Y.Kurylev, M.Lassas and M.Salo, The Calderón problem in transversally anisotropic geometries, J. Eur. Math. Soc.18(11) (2016), 2579–2626. doi:10.4171/jems/649.
9.
A.Feizmohammadi, J.Ilmavirta, Y.Kian and L.Oksanen, Recovery of time dependent coefficients from boundary data for hyperbolic equations, J. Spectr. Theory11 (2021), 1107–1143. doi:10.4171/jst/367.
10.
D.Gilbarg and N.Trudinger, Elliptic Partial Differential Equations of Second Order, revised third edn, Springer-Verlag, Berlin, 2001.
11.
C.Guillarmou, Lens rigidity for manifolds with hyperbolic trapped sets, J. Amer. Math. Soc.30(2) (2017), 561–599. doi:10.1090/jams/865.
12.
B.Haberman, Unique determination of a magnetic Schrödinger operator with unbounded magnetic potential from boundary data, Int. Math. Res. Not.2018(4) (2018), 1080–1128.
13.
C.Kenig and M.Salo, The Calderón problem with partial data on manifolds and applications, Anal. PDE6(8) (2013), 2003–2048. doi:10.2140/apde.2013.6.2003.
14.
C.Kenig and M.Salo, Recent progress in the Calderón problem with partial data, Contemp. Math.615 (2014), 193–222. doi:10.1090/conm/615/12245.
15.
C.E.Kenig, J.Sjöstrand and G.Uhlmann, The Calderón problem with partial data, Ann. of Math.165(2) (2007), 567–591. doi:10.4007/annals.2007.165.567.
16.
K.Knudsen and M.Salo, Determining nonsmooth first order terms from partial boundary measurements, Inverse Probl. Imaging1(2) (2007), 349–369. doi:10.3934/ipi.2007.1.349.
17.
K.Krupchyk and G.Uhlmann, Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential, Commun. Math. Phys.327(3) (2014), 993–1009. doi:10.1007/s00220-014-1942-z.
18.
K.Krupchyk and G.Uhlmann, Inverse problems for advection diffusion equations in admissible geometries, Commun. Partial. Differ. Equ.43(4) (2018), 585–615. doi:10.1080/03605302.2018.1446163.
19.
K.Krupchyk and G.Uhlmann, Inverse problems for magnetic Schrödinger operators in transversally anisotropic geometries, Commun. Math. Phys.361(2) (2018), 525–582. doi:10.1007/s00220-018-3182-0.
20.
G.Nakamura, Z.Sun and G.Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann.303 (1995), 377–388. doi:10.1007/BF01460996.
21.
A.Panchenko, An inverse problem for the magnetic Schrödinger equation and quasi-exponential solutions of nonsmooth partial differential equations, Inverse Problems18(5) (2002), 1421–1434. doi:10.1088/0266-5611/18/5/314.
22.
M.Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, volume 139, Ann. Acad. Sci. Fenn. Math. Diss. (2004).
23.
S.Selim, Partial data inverse problems for magnetic Schrödinger operators with potentials of low regularity, 2022, arXiv preprint arXiv:2210.06595.
24.
J.Sjöstrand, Weyl law for semi-classical resonances with randomly perturbed potentials, Mém. Soc. Math. Fr. (N. S.) (2014).
25.
Z.Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc.338(2) (1993), 953–969. doi:10.1090/S0002-9947-1993-1179400-1.
26.
J.Sylvester and G.Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math.125(1) (1987), 153–169. doi:10.2307/1971291.
27.
C.Tolmasky, Exponentially growing solutions for nonsmooth first-order perturbations of the Laplacian, SIAM J. Math. Anal.29(1) (1998), 116–133. doi:10.1137/S0036141096301038.
