Identification of a Connection from Cauchy Data on a Riemann Surface with Boundary

Guillarmou, Colin; Tzou, Leo

doi:10.1007/s00039-011-0110-2

Cited by 34 publications

(70 citation statements)

References 25 publications

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“…It also implies unique determination of a potential from the fixed energy scattering amplitude in two dimensions. A general result for first order systems is in [3], generalizing the results of [71] and [104].…”

Section: Bukhgeim's Resultsmentioning

confidence: 68%

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Inverse problems: seeing the unseen

2014

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This survey article deals mainly with two inverse problems and the relation between them. The first inverse problem we consider is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This is called electrical impedance tomography and also Calderón's problem since the famous analyst proposed it in the mathematical literature (Calderón in On an inverse boundary value problem. Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro, 1980), Soc Brasil Mat. Rio de Janeiro, pp. [65][66][67][68][69][70][71][72][73] 1980). The second is on travel time tomography. The question is whether one can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as a geometry problem, the boundary rigidity problem. Can we determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points? These two inverse problems concern visibility, that is whether we can determine the internal properties of a medium by making measurements at the boundary. The last topic of this paper considers the opposite issue: invisibility: Can one make objects invisible to different types of waves, including light?Communicated by Ari Laptev.

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Section: Bukhgeim's Resultsmentioning

confidence: 68%

“…Rio de Janeiro, pp. [65][66][67][68][69][70][71][72][73] 1980). The second is on travel time tomography.…”

mentioning

confidence: 99%

Inverse problems: seeing the unseen

2014

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“…Finally, assuming that q (1) = q (2) , using parallel transport along loops in M , boundary reconstruction of the magnetic potential, and unique continuation arguments, as in [14] and [5], we show that the fluxes of the magnetic potentials A (1) and A (2) satisfy…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 85%

“…In doing so we shall follow [14], [5], and replace γ by a homotopically equivalent loop γ : [0, 2] → M such that γ(0) = m, γ(1) = m ∈ ∂M , m = m, γ 1 := γ((0, 1)) ⊂ M 0 and γ 2 := γ([1, 2]) ⊂ ∂M . As explained in [5], we may assume that γ 1 and γ 2 are embedded curves.…”

mentioning

confidence: 99%

Inverse Problems for Magnetic Schrödinger Operators in Transversally Anisotropic Geometries

Krupchyk

Uhlmann

2018

Commun. Math. Phys.

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We study inverse boundary problems for magnetic Schrödinger operators on a compact Riemannian manifold with boundary of dimension ≥ 3. In the first part of the paper we are concerned with the case of admissible geometries, i.e. compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Cauchy data on the boundary of the manifold for the magnetic Schrödinger operator with L ∞ magnetic and electric potentials, determines the magnetic field and electric potential uniquely.In the second part of the paper we address the case of more general conformally transversally anisotropic geometries, i.e. compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a compact manifold, which need not be simple. Here, under the assumption that the geodesic ray transform on the transversal manifold is injective, we prove that the knowledge of the Cauchy data on the boundary of the manifold for a magnetic Schrödinger operator with continuous potentials, determines the magnetic field uniquely. Assuming that the electric potential is known, we show that the Cauchy data determines the magnetic potential up to a gauge equivalence.

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“…Another additional feature which we must consider is the uniformity of the estimates in the stationary phase expansion. As such we must construct slightly different phase functions than in [7] to be used in these solutions and refine several estimates from mentioned works. This work is done in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Stability for a magnetic Schrödinger operator on a Riemann surface with boundary

Andersson¹,

Tzou²

2018

Inverse Problems &Amp; Imaging

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We consider a magnetic Schrödinger operator (∇ X ) * ∇ X + q on a compact Riemann surface with boundary and prove a log log-type stability estimate in terms of Cauchy data for the electric potential and magnetic field under the assumption that they satisfy appropriate a priori bounds. We also give a similar stability result for the holonomy of the connection 1-form X. Cauchy-Riemann operators on MNext we will introduce the Cauchy Riemann operators ∂,∂ as mappings of kforms to (k + 1)-forms, k = 0, 1.On functions their action is defined by ∂f := π 1,0 df,∂f := π 0,1 df.In holomorphic coordinates this is simply ∂f := ∂ z f dz,∂f := ∂zf dz,

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Identification of a Connection from Cauchy Data on a Riemann Surface with Boundary

Cited by 34 publications

References 25 publications

Inverse problems: seeing the unseen

Inverse problems: seeing the unseen

Inverse Problems for Magnetic Schrödinger Operators in Transversally Anisotropic Geometries

Stability for a magnetic Schrödinger operator on a Riemann surface with boundary

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