Inverse problems for Schrödinger equations with Yang–Mills potentials in domains with obstacles and the Aharonov–Bohm effect

Journal of Physics Conference Series 04/2005; 12(1). DOI: 10.1088/1742-6596/12/1/003
Source: arXiv


We study the inverse boundary value problems for the Schr\"{o}dinger
equations with Yang-Mills potentials in a bounded domain $\Omega_0\subset\R^n$
containing finite number of smooth obstacles $\Omega_j,1\leq j \leq r$. We
prove that the Dirichlet-to-Neumann operator on $\partial\Omega_0$ determines
the gauge equivalence class of the Yang-Mills potentials. We also prove that
the metric tensor can be recovered up to a diffeomorphism that is identity on

  • Source
    • "More recently, the injectivity result for the non-abelian Radon transform was extended to any simply connected surface with strictly convex boundary and no conjugate points [30] and to higher dimensions and negative curvature [17]. There is a result due to G. Eskin [11] that implies Corollary 1.2 under the assumption that M is a domain in Euclidean space with obstacles. Our proof seems however simpler. "
    [Show abstract] [Hide abstract]
    ABSTRACT: We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a corollary we derive an elliptic analogue of the main result which solves a Calderon problem for connections on a cylinder.
    Full-text · Article · Sep 2015
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Magnetic Aharonov-Bohm effect (AB effect) was studied in hundreds of papers starting with the seminal paper of Aharonov and Bohm [AB] published in 1959. We give a new proof of the magnetic Aharonov-Bohm effect without using the scattering theory and the theory of inverse boundary value problems. We consider separately the cases of one and several obstacles. The electric AB effect was studied much less. We give the first proof of the electric AB effect in domains with moving boundaries. When the boundary does not move with the time the electric AB effect is absent.
    Preview · Article · Aug 2013 · Communications in Mathematical Physics
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We present a modification of the BC-method in the inverse hyperbolic problems. The main novelty is the study of the restrictions of the solutions to the characteristic surfaces instead of the fixed time hyperplanes. The main result is that the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the self-adjoint hyperbolic operator up to a diffeomorphism and a gauge transformation. In this paper we prove the crucial local step. The global step of the proof will be presented in the forthcoming paper.
    Preview · Article · May 2006 · Inverse Problems
Show more