Inverse problems and index formulae for Dirac operators

Anderson, 2004, Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand inverse boundary problem, Invent. Math., 105, 261, 10.1007/s00222-004-0371-6 Astala, 2006, Calderon's inverse conductivity problem in the plane, Ann. of Math., 163, 265, 10.4007/annals.2006.163.265 Astala, 2005, Calderon's inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30, 207, 10.1081/PDE-200044485 Atiyah, 1975, Spectral asymmetry and Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc., 77, 43, 10.1017/S0305004100049410 Belishev, 1987, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297, 524 Belishev, 1987, A nonstationary inverse problem for the multidimensional wave equation “in the large”, Zap. Nauchn. Sem. (LOMI), 165, 21 Belishev, 1992, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17, 767, 10.1080/03605309208820863 Berline, 1992 Bismut, 1990, Families index for manifolds with boundary, superconnections, and cones. I. Families of manifolds with boundary and Dirac operators, J. Funct. Anal., 89, 313, 10.1016/0022-1236(90)90098-6 Blagovestchenskii, 1969, A one-dimensional inverse boundary value problem for a second order hyperbolic equation, Zap. Nauchn. Sem. LOMI, 15, 85 Booss-Bavnbek, 1993 Brüning, 1990, L2-index theorems on certain complete manifolds, J. Differential Geom., 32, 491, 10.4310/jdg/1214445317 Brüning, 2001, On boundary value problems for Dirac type operators. I. Regularity and self-adjointness, J. Funct. Anal., 185, 1, 10.1006/jfan.2001.3753 Daskalov, 2000, Explicit formulae for the inverse problem for the regular Dirac operator, Inverse Problems, 16, 247, 10.1088/0266-5611/16/1/318 Eskin, 2003, Inverse boundary value problems and the Aharonov–Bohm effect, Inverse Problems, 19, 49, 10.1088/0266-5611/19/1/303 Farinelli, 1998, On the spectrum of the Dirac operator under boundary conditions, J. Geom. Phys., 28, 67, 10.1016/S0393-0440(98)00013-8 Gilbarg, 1983 Gilkey, 1973, Curvature and the eigenvalues of the Laplacian for elliptic complexes, Adv. Math., 10, 344, 10.1016/0001-8708(73)90119-9 Gilkey, 1993, On the index of geometrical operators for Riemannian manifolds with boundary, Adv. Math., 102, 129, 10.1006/aima.1993.1063 Grebert, 1992, Inverse scattering for the Dirac operator on the real line, Inverse Problems, 8, 787, 10.1088/0266-5611/8/5/007 Greenleaf, 2007, Full-wave invisibility of active devices at all frequencies, Comm. Math. Phys., 275, 749, 10.1007/s00220-007-0311-6 Greenleaf, 2009, Invisibility and inverse problems, Bull. Amer. Math. Soc. (N.S.), 46, 55, 10.1090/S0273-0979-08-01232-9 Hachem, 1995, The ∂¯ approach to inverse scattering for Dirac operators, Inverse Problems, 11, 123, 10.1088/0266-5611/11/1/007 Hijazi, 2001, Eigenvalues of the Dirac operator on manifolds with boundary, Comm. Math. Phys., 221, 255, 10.1007/s002200100475 Hijazi, 2002, Eigenvalue boundary problems for the Dirac operator, Comm. Math. Phys., 231, 375, 10.1007/s00220-002-0725-0 Horvath, 2001, On the inverse spectral theory of Schrödinger and Dirac operators, Trans. Amer. Math. Soc., 353, 4155, 10.1090/S0002-9947-01-02765-9 Imanuvilov, 2001, Global uniqueness and stability in determining coefficients of wave equations, Comm. Partial Differential Equations, 26, 1409, 10.1081/PDE-100106139 Isozaki, 1997, Inverse scattering theory for Dirac operators, Ann. Inst. H. Poincaré Phys. Theor., 66, 237 Jung, 1997, Geometrical approach to inverse scattering for the Dirac equation, J. Math. Phys., 38, 39, 10.1063/1.531856 Kachalov, 1998, Multidimensional inverse problem with incomplete boundary spectral data, Comm. Partial Differential Equations, 23, 55 Katchalov, 2001, Inverse Boundary Spectral Problems, vol. 123 Katchalov, 2004, Equivalence of time-domain inverse problems and boundary spectral problems, Inverse Problems, 20, 419, 10.1088/0266-5611/20/2/007 Kawashita, 2000, Harmonic moments and an inverse problem for the heat equation, SIAM J. Math. Anal., 32, 522, 10.1137/S0036141099353035 Krupchyk, 2008, Inverse spectral problems on a closed manifold, J. Math. Pures Appl., 90, 42, 10.1016/j.matpur.2008.02.009 Kurylev, 1992, Admissible groups of transformations that preserve the boundary spectral data in multidimensional inverse problems, Dokl. Akad. Nauk, 327, 322 Kurylev, 1995, An inverse boundary problem for the Schrödinger operator with magnetic field, J. Math. Phys., 36, 2761, 10.1063/1.531064 Kurylev, 1997, Multidimensional Gel'fand inverse problem and boundary distance map, 1 Kurylev, 2000, Gelf'and inverse problem for a quadratic operator pencil, J. Funct. Anal., 176, 247, 10.1006/jfan.2000.3615 Kurylev, 2003, Reconstruction of manifold from electromagnetic boundary measurements, vol. 333, 147 Kurylev, 2004, Focusing waves in electromagnetic inverse problems, vol. 348, 11 Kurylev, 2006, Maxwell's equations with a polarization independent wave velocity: Direct and inverse problems, J. Math. Pures Appl., 86, 237, 10.1016/j.matpur.2006.01.008 Lasiecka, 2000 Lassas, 2001, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sci. Ecole Norm. Sup., 34, 771, 10.1016/S0012-9593(01)01076-X McKean, 1967, Curvature and the eigenvalues of the Laplacian, J. Differential Geom., 1, 43, 10.4310/jdg/1214427880 Melrose, 1993 Müller, 1996, On the L2-index of Dirac operators on manifolds with corners of codimension two, I, J. Differential Geom., 44, 97, 10.4310/jdg/1214458741 Nachman, 1988, Reconstructions from boundary measurements, Ann. of Math., 128, 531, 10.2307/1971435 Nachman, 1988, An n-dimensional Borg–Levinson theorem, Comm. Math. Phys., 115, 595, 10.1007/BF01224129 Nakamura, 2000, Uniqueness for an inverse boundary value problem for Dirac operators, Comm. Partial Differential Equations, 25, 1327 Nogami, 1986, Dirac equation with a separable potential, Phys. Rev. C, 34, 1855, 10.1103/PhysRevC.34.1855 Pestov, 2005, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math., 161, 1093, 10.4007/annals.2005.161.1093 Russell, 1978, Controllability and stabilizability theory for linear partial differential equations, SIAM Rev., 20, 639, 10.1137/1020095 M. Salo, L. Tzou, Carleman estimates and inverse problems for Dirac operators, Math. Ann., in press Seeley, 1964, Extension of C∞-functions defined in a half space, Proc. Amer. Math. Soc., 15, 625 Sylvester, 1987, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125, 153, 10.2307/1971291 Tataru, 1995, Unique continuation for solutions to PDEs, between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20, 855, 10.1080/03605309508821117 Tataru, 1999, Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl., 78, 505, 10.1016/S0021-7824(99)00016-1 Taylor, 1996 Tsuchida, 1998, An inverse boundary value problem for Dirac operators with small potentials, Kyushu J. Math., 52, 361, 10.2206/kyushujm.52.361 Weder, 2002, The Aharonov–Bohm effect and time-dependent inverse scattering theory, Inverse Problems, 18, 1041, 10.1088/0266-5611/18/4/307 Wilcox, 1964, The domain of dependence inequality for symmetric hyperbolic systems, Bull. Amer. Math. Soc., 70, 149, 10.1090/S0002-9904-1964-11056-9