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12
Recovering Asymptotics Of Metrics From Fixed Energy Scattering Data
"... The problem of recovering the asymptotics of a short range perturbation of the Euclidean metric on R from fixed energy scattering data is studied. It is shown that if two such metrics, g 1 ; g 2 ; have scattering data at some fixed energy which are equal up to smoothing, then there exists a dif ..."
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Cited by 16 (6 self)
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The problem of recovering the asymptotics of a short range perturbation of the Euclidean metric on R from fixed energy scattering data is studied. It is shown that if two such metrics, g 1 ; g 2 ; have scattering data at some fixed energy which are equal up to smoothing, then there exists a diffeomorphism / `fixing infinity' such that / g 1 \Gammag 2 is rapidly decreasing. Given the scattering matrix at two energies, it is shown that the asymptotics of a metric and a short range potential can be determined simultaneously.
Boundary Regularity for the Ricci Equation, Geometric Convergence, and Gel'fand's Inverse Boundary Problem
"... Abstract This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The secon ..."
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Cited by 14 (13 self)
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Abstract This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand, making essential use of the results of the first two parts. 1.
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
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RIGIDITY OF BROKEN GEODESIC FLOW AND INVERSE PROBLEMS
, 2008
"... Consider broken geodesics α([0, l]) on a compact Riemannian manifold (M, g) with boundary of dimension n ≥ 3. The broken geodesics are unions of two geodesics with the property that they have a common end point. Assume that for every broken geodesic α([0, l]) starting at and ending to the boundary ∂ ..."
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Cited by 3 (0 self)
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Consider broken geodesics α([0, l]) on a compact Riemannian manifold (M, g) with boundary of dimension n ≥ 3. The broken geodesics are unions of two geodesics with the property that they have a common end point. Assume that for every broken geodesic α([0, l]) starting at and ending to the boundary ∂M we know the starting point and direction (α(0), α ′ (0)), the end point and direction (α(l), α ′ (l)), and the length l. We show that this data determines uniquely, up to an isometry, the manifold (M, g). This result has applications in inverse problems on very heterogeneous media for situations where there are many scattering points in the medium, and arises in several applications including geophysics and medical imaging. As an example we consider the inverse problem for the radiative transfer equation (or the linear transport equation) with a nonconstant wave speed. Assuming that the scattering kernel is everywhere positive, we show that the boundary measurements determine the wave speed inside the domain up to an isometry.
Dynamical inverse problem for the Schrödinger equation (the BCmethod)
, 2002
"... this paper we use important results by R. Triggiani and P.F. Yao [12] on the exact boundary controllability of the Schrodinger system ..."
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Cited by 3 (1 self)
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this paper we use important results by R. Triggiani and P.F. Yao [12] on the exact boundary controllability of the Schrodinger system
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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Cited by 1 (1 self)
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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
Gauge Equivalence and Inverse Scattering for AharonovBohm Effect, Preprint ArXiv: mathph 0809.3291
"... Abstract. We consider the AharonovBohm effect for the Schrödinger operator H = (−i∇x − A(x)) 2 +V (x) and the related inverse problem in an exterior domain Ω in R 2 with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering o ..."
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Cited by 1 (0 self)
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Abstract. We consider the AharonovBohm effect for the Schrödinger operator H = (−i∇x − A(x)) 2 +V (x) and the related inverse problem in an exterior domain Ω in R 2 with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering operator determines the domain Ω and H up to gauge equivalence under the equal flux condition. We also show that the flux is determined by the scattering operator if the obstacle Ω c is convex. 1.
Stability and Reconstruction in Gel’fand Inverse Boundary Spectral Problem
, 2008
"... Abstract. We consider stability and approximate reconstruction of Riemannian manifold when the finite number of eigenvalues of the LaplaceBeltrami operator and the boundary values of the corresponding eigenfunctions are given. The reconstruction can be done in stable way when manifold is a priori k ..."
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Abstract. We consider stability and approximate reconstruction of Riemannian manifold when the finite number of eigenvalues of the LaplaceBeltrami operator and the boundary values of the corresponding eigenfunctions are given. The reconstruction can be done in stable way when manifold is a priori known to satisfy natural geometrical conditions related to curvature and other invariant quantities. I. Introduction In this paper we consider the questions of stability and approximate reconstruction in the inverse boundary spectral problem which is also called the generalized Gelfand inverse problem [13] for Riemannian manifolds. To formulate the problem and the main results, we need to introduce some basic notations. We will denote by (M,g) an unknown,