Results 1  10
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10
DETERMINING NONSMOOTH FIRST ORDER TERMS FROM PARTIAL BOUNDARY MEASUREMENTS
, 2006
"... Abstract. We extend results of Dos Santos FerreiraKenigSjöstrandUhlmann (arXiv:math.AP/0601466) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schrödinger operator determine uniquely the magnetic field related to a Hölder continuous potential. ..."
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Cited by 12 (6 self)
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Abstract. We extend results of Dos Santos FerreiraKenigSjöstrandUhlmann (arXiv:math.AP/0601466) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schrödinger operator determine uniquely the magnetic field related to a Hölder continuous potential. We give a similar result for determining a convection term. The proofs involve Carleman estimates, a smoothing procedure, and an extension of the NakamuraUhlmann pseudodifferential conjugation method to logarithmic Carleman weights. 1.
Stability estimates for coefficients of magnetic Schrödinger . . .
, 2008
"... In this paper we establish a log logtype estimate which shows that in dimension n ≥ 3 the magnetic field and the electric potential of the magnetic Schrödinger equation depends stably on the Dirichlet to Neumann (DN) map even when the boundary measurement is taken only on a subset that is slightly ..."
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Cited by 8 (0 self)
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In this paper we establish a log logtype estimate which shows that in dimension n ≥ 3 the magnetic field and the electric potential of the magnetic Schrödinger equation depends stably on the Dirichlet to Neumann (DN) map even when the boundary measurement is taken only on a subset that is slightly larger than half of the boundary ∂Ω. Furthermore, we prove that in the case when the measurement is taken on all of ∂Ω one can establish a better estimate that is of logtype. The proofs involve the use of the complex geometric optics (CGO) solutions of the magnetic Schrödinger equation constructed in [8] then follow a similar line of argument as in [1]. In the partial data estimate we follow the general strategy of [5] by using the Carleman estimate established in [4] and a continuous dependence result for analytic continuation developed in [14].
Inverse boundary value problem for Maxwell equations with local data ∗†
, 902
"... We prove a uniqueness theorem for an inverse boundary value problem for the Maxwell system with boundary data assumed known only in part of the boundary. We assume that the inaccessible part of the boundary is either part of a plane, or part of a sphere. This work generalizes the results obtained by ..."
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Cited by 5 (3 self)
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We prove a uniqueness theorem for an inverse boundary value problem for the Maxwell system with boundary data assumed known only in part of the boundary. We assume that the inaccessible part of the boundary is either part of a plane, or part of a sphere. This work generalizes the results obtained by Isakov [I] for the Schrödinger equation to Maxwell equations. Introduction. Let Ω ⊂ R 3 be a bounded domain with C 1,1 boundary, and let ε, µ, σ be C 2 functions in Ω (ε is the permittivity, µ the permeability, and σ the conductivity). We will assume that the coefficients satisfy the positivity conditions γ = ε + iσ/ω, ε> 0, µ> 0, σ ≥ 0 in Ω. (0.1) Let D = −i∇, let ν be the exterior unit normal to ∂Ω, and consider the timeharmonic
CARLEMAN ESTIMATES AND INVERSE PROBLEMS FOR DIRAC OPERATORS
, 709
"... Abstract. We consider limiting Carleman weights for Dirac operators and prove corresponding Carleman estimates. In particular, we show that limiting Carleman weights for the Laplacian also serve as limiting weights for Dirac operators. As an application we consider the inverse problem of recovering ..."
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Cited by 4 (4 self)
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Abstract. We consider limiting Carleman weights for Dirac operators and prove corresponding Carleman estimates. In particular, we show that limiting Carleman weights for the Laplacian also serve as limiting weights for Dirac operators. As an application we consider the inverse problem of recovering a Lipschitz continuous magnetic field and electric potential from boundary measurements for the Pauli Dirac operator. 1.
MULTIDIMENSIONAL VERSIONS OF A DETERMINANT FORMULA DUE TO JOST AND PAIS
"... Abstract. We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schrödinger operator on a halfline to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrö ..."
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Cited by 2 (0 self)
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Abstract. We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schrödinger operator on a halfline to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation, generalizes to higher dimensions. In this multidimensional extension, the halfline is replaced by an open set Ω ⊂ Rn, n = 2, 3, where Ω has a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on ∂Ω and invokes the corresponding DirichlettoNeumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants perturbation associated with operators in L2 (Ω; dnx) to modified Fredholm determinants associated with operators in L2 (∂Ω; dn−1σ), n = 2, 3. 1.
Stability estimate in an inverse problem for non autonomous magnetic Schrödinger equations
, 2012
"... ..."
DETERMINING A MAGNETIC SCHRÖDINGER OPERATOR FROM PARTIAL CAUCHY DATA
, 2006
"... Abstract. In this paper we show, in dimension n ≥ 3, that knowledge of the Cauchy data for the Schrödinger equation in the presence of a magnetic potential, measured on possibly very small subsets of the boundary, determines uniquely the magnetic field and the electric potential. We follow the gener ..."
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Abstract. In this paper we show, in dimension n ≥ 3, that knowledge of the Cauchy data for the Schrödinger equation in the presence of a magnetic potential, measured on possibly very small subsets of the boundary, determines uniquely the magnetic field and the electric potential. We follow the general strategy of [7] using a richer set of solutions to the Dirichlet problem that has been used in previous works on this problem. 1.
s1
, 705
"... Abstract. We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schrödinger operator on a halfline to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrö ..."
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Abstract. We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schrödinger operator on a halfline to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation, generalizes to higher dimensions. In this multidimensional extension the halfline is replaced by an open set Ω ⊂ R n, n ∈ N, n ≥ 2, where Ω has a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on ∂Ω and invokes the corresponding DirichlettoNeumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants associated with operators in L 2 (Ω; d n x), n ∈ N, to modified Fredholm determinants associated with operators in L 2 (∂Ω; d n−1 σ), n ≥ 2. Applications involving the Birman–Schwinger principle and eigenvalue counting functions are discussed. 1.
INVERSE SCATTERING FOR THE MAGNETIC SCHRÖDINGER OPERATOR
, 908
"... Abstract. We show that fixed energy scattering measurements for the magnetic Schrödinger operator uniquely determine the magnetic field and electric potential in dimensions n ≥ 3. The magnetic potential, its first derivatives, and the electric potential are assumed to be exponentially decaying. This ..."
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Abstract. We show that fixed energy scattering measurements for the magnetic Schrödinger operator uniquely determine the magnetic field and electric potential in dimensions n ≥ 3. The magnetic potential, its first derivatives, and the electric potential are assumed to be exponentially decaying. This improves an earlier result of Eskin and Ralston [4] which considered potentials with many derivatives. The proof is close to arguments in inverse boundary problems, and is based on constructing complex geometrical optics solutions to the Schrödinger equation via a pseudodifferential conjugation argument. 1.
RECONSTRUCTIONS FROM BOUNDARY MEASUREMENTS ON ADMISSIBLE MANIFOLDS
"... Abstract. We prove that a potential q can be reconstructed from the DirichlettoNeumann map for the Schrödinger operator −∆g + q in a fixed admissible 3dimensional Riemannian manifold (M, g). We also show that an admissible metric g in a fixed conformal class can be constructed from the Dirichlet ..."
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Abstract. We prove that a potential q can be reconstructed from the DirichlettoNeumann map for the Schrödinger operator −∆g + q in a fixed admissible 3dimensional Riemannian manifold (M, g). We also show that an admissible metric g in a fixed conformal class can be constructed from the DirichlettoNeumann map for ∆g. This is a constructive version of earlier uniqueness results by Dos Santos Ferreira et al. [7] on admissible manifolds, and extends the reconstruction procedure of Nachman [21] in Euclidean space. The main points are the derivation of a boundary integral equation characterizing the boundary values of complex geometrical optics solutions, and the development of associated layer potentials adapted to a cylindrical geometry. 1.