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10
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
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Solving inverse problems with spectral data, Surveys on Solution Methods for Inverse Problems
 MR 2001f:35431
, 2000
"... We consider a two dimensional membrane. The goal is to find properties of the membrane or properties of a force on the membrane. The data is natural frequencies or mode shape measurements. As a result, the functional relationship between the data and the solution of our inverse problem is both indir ..."
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We consider a two dimensional membrane. The goal is to find properties of the membrane or properties of a force on the membrane. The data is natural frequencies or mode shape measurements. As a result, the functional relationship between the data and the solution of our inverse problem is both indirect and nonlinear. In this paper we describe three distinct approaches to this problem. In the first approach the data is mode shape level sets and frequencies. Here formulas for approximate solutions are given based on perturbation results. In the second approach the data is frequencies and boundary mode shape measurements; uniqueness results are obtained using the boundary control method. In the third approach the data is frequencies for four boundary value problems. Local existence, uniqueness results are established together with numerical results for approximate solutions. 2Introduction We consider two dimensional membranes. Spectral data is measured for these
Highvelocity estimates for the scattering operator and the AharonovBohm effect in three dimensions
 Commun. in Math. Phys
"... We obtain highvelocity estimates with error bounds for the scattering operator of the Schrödinger equation in three dimensions with electromagnetic potentials in the exterior of bounded obstacles that are handlebodies. A particular case is a finite number of tori. We prove our results with timedep ..."
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We obtain highvelocity estimates with error bounds for the scattering operator of the Schrödinger equation in three dimensions with electromagnetic potentials in the exterior of bounded obstacles that are handlebodies. A particular case is a finite number of tori. We prove our results with timedependent methods. We consider highvelocity estimates where the direction of the velocity of the incoming electrons is kept fixed as its absolute value goes to infinity. In the case of one torus our results give a rigorous proof that quantum mechanics predicts the interference patterns observed in the fundamental experiments of Tonomura et al. that gave a conclusive evidence of the existence of the AharonovBohm effect using a toroidal magnet. We give a method for the reconstruction of the flux of the magnetic field over a crosssection of the torus modulo 2π. Equivalently, we determine modulo 2π the difference in phase for two electrons that travel to infinity, when one goes inside the hole and the other outside it. For this purpose we only need the highvelocity limit of the scattering operator for one direction of the velocity of the incoming electrons. When there are several torior more generally handlebodiesthe information that we obtain in the fluxes, and on the difference of phases, depends on the relative position of the tori and on the direction of the velocities when we take the highvelocity limit of the incoming electrons. For some locations of the tori we can determine all the fluxes modulo 2π by taking the highvelocity limit in only one direction. We also give a method for the unique reconstruction of the electric potential and the magnetic field outside the handlebodies from the highvelocity limit of the scattering operator.
Multidimensional Gel’fand inverse boundary spectral problem: uniqueness and stability
 Cubo
"... Here we consider the Gel'fand Inverse Spectral Boundary Problem. Let us start with a nonrigorous introduction to this class of problems. Assume we have a manifold with boundary (M; @M), a vector bundle over M, and a linear elliptic dierential operator A acting on smooth sections of which are ..."
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Cited by 2 (2 self)
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Here we consider the Gel'fand Inverse Spectral Boundary Problem. Let us start with a nonrigorous introduction to this class of problems. Assume we have a manifold with boundary (M; @M), a vector bundle over M, and a linear elliptic dierential operator A acting on smooth sections of which are denoted
INVERSE SPECTRAL PROBLEMS ON A CLOSED MANIFOLD
, 709
"... Abstract. In this paper we consider two inverse problems on a closed connected Riemannian manifold (M, g). The first one is a direct analog of the Gel’fand inverse boundary spectral problem. To formulate it, assume that M is divided by a hypersurface Σ into two components and we know the eigenvalues ..."
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Abstract. In this paper we consider two inverse problems on a closed connected Riemannian manifold (M, g). The first one is a direct analog of the Gel’fand inverse boundary spectral problem. To formulate it, assume that M is divided by a hypersurface Σ into two components and we know the eigenvalues λj of the Laplace operator on (M, g) and also the Cauchy data, on Σ, of the corresponding eigenfunctions φj, i.e. φjΣ, ∂νφjΣ, where ν is the normal to Σ. We prove that these data determine (M, g) uniquely, i.e. up to an isometry. In the second problem we are given much less data, namely, λj and φjΣ only. However, if Σ consists of at least two components, Σ1, Σ2, we are still able to determine (M, g) assuming some conditions on M and Σ. These conditions are formulated in terms of the spectra of the manifolds with boundary obtained by cutting M along Σi, i = 1, 2, and are of a generic nature. We consider also some other inverse problems on M related to the above with data which is easier to obtain from measurements than the spectral data described. 1. Introduction and
Inverse Boundary Spectral Problem for Riemannian Polyhedra
, 2008
"... We consider an admissible Riemannian polyhedron with piecewise smooth boundary. The associated Laplace defines the boundary spectral data as the set of eigenvalues and restrictions to the boundary of the corresponding eigenfunctions. In this paper we prove that the boundary spectral data prescribed ..."
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We consider an admissible Riemannian polyhedron with piecewise smooth boundary. The associated Laplace defines the boundary spectral data as the set of eigenvalues and restrictions to the boundary of the corresponding eigenfunctions. In this paper we prove that the boundary spectral data prescribed on an open subset of the polyhedron boundary determine the admissible Riemannian polyhedron uniquely. 1
Maxwell’s Equations with Scalar Impedance: Direct and Inverse Problems
, 2002
"... Abstract: The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell’s equations written for differential forms over a 3manifold are analysed. The system is extended to a Dirac type first order elliptic system on the Grassmannian bundle ove ..."
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Abstract: The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell’s equations written for differential forms over a 3manifold are analysed. The system is extended to a Dirac type first order elliptic system on the Grassmannian bundle over the manifold. The second part of the article deals with the dynamical inverse boundary value problem of determining the electromagnetic material parameters from boundary measurements. By using the boundary control method, it is proved that the dynamical boundary data determines the electromagnetic travel time metric as well as the scalar wave impedance on the manifold. This invariant result leads also to a complete characterization of the nonuniqueness of the corresponding inverse problem in bounded domains of R 3.
Equivalence of timedomain inverse problems and boundary spectral problems
, 2008
"... Abstract. We consider inverse problems for wave, heat and Schrödingertype operators and corresponding spectral problems on domains of R n and compact manifolds. Also, we study inverse problems where coefficients of partial differential operator have to be found when one knows how much energy it is r ..."
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Abstract. We consider inverse problems for wave, heat and Schrödingertype operators and corresponding spectral problems on domains of R n and compact manifolds. Also, we study inverse problems where coefficients of partial differential operator have to be found when one knows how much energy it is required to force the solution to have given boundary values, i.e., one knows how much energy is needed to make given measurements. The main result of the paper is to show that all these problems are shown to be equivalent. 1
Wave imaging
, 2010
"... This chapter discusses imaging methods related to wave phenomena, and in particular inverse problems for the wave equation will be considered. The first part of the chapter explains the boundary control method for determining a wave speed of a medium from the response operator which models boundary ..."
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This chapter discusses imaging methods related to wave phenomena, and in particular inverse problems for the wave equation will be considered. The first part of the chapter explains the boundary control method for determining a wave speed of a medium from the response operator which models boundary measurements. The second part discusses the scattering relation and travel times, which are different types of boundary data contained in the response operator. The third part gives a brief introduction to curvelets in wave imaging for media with nonsmooth wave speeds. The focus will be on theoretical results and methods.