Boundary control in reconstruction of manifolds and metrics (the BC method)

Belishev, M. I.

doi:10.1088/0266-5611/13/5/002

Cited by 201 publications

(273 citation statements)

References 33 publications

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“…In particular, Sa Barreto [SaBa05] proved that the coincidence of the scattering operators gives rise to an isometry of associated metrics. Here the essential role is played by the boundary control method presented by Belishev [Be87], (see also [BeKu87], [Be97], [BeKu92]), which makes it possible to reconstruct a Riemannian manifold from the boundary spectral data of the associated Laplace-Belrami operator. A feature of Melrose theory is that it proves the analytic continuation of the resolvent of Laplace-Beltrami operator for a broad class of metric so that it enables us to study the resonance, another important subject in spectral and scattering theory ( [GuZw97]), [Zw99]).…”

Section: Spectral and Scattering Theory On Hyperbolic Manifoldsmentioning

confidence: 99%

Introduction to Spectral Theory and Inverse Problem on Asymptotically Hyperbolic Manifolds

Isozaki¹,

Kurylev²

2014

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Since −∆e ix·ξ = |ξ| 2 e ix·ξ , e ix·ξ is an eigenfunction of −∆. Therefore (0.1) and (0.2) illustrate the expansion of arbitrary functions in terms of eigenfunctions (more appropriately generalized eigenfunctions since they do not belong to L 2 (R n )) of the Laplacian.There are two directions of development of the above fact. One is quantum mechanics, where the Schrödinger operator H = −∆ + V (x) is the most basic tool to decribe the physical system of atoms or molecules. If H has the continuous spectrum, it is known that there exists a system of generalized eigenfunctions of H which play the same role as e ix·ξ . Moreover, by using these generalized eigenfunctions one can define an operator called the scattering matrix or the S-matrix, which is the fundamental object to study the physical properties of quantum mechanical particles through the scattering experiment.The other direction is the Fourier transform on manifolds, especially on homogeneous spaces of Lie groups, which is a central theme in the representation theory. Hyperbolic manifolds, one of the deepest sources of classical mathematics, appear also in this context. In particular, hyperbolic quotient manifolds by the action of discrete subgroups of SL(2, R) and the associated S-matrix are important objects in number theory. 0.2. Perturbation of the continuous spectrum. The aim of the perturbation theory of continuous spectrum is, given an operator H 0 whose spectral property is rather easy to understand, to study the spectral properties of H 0 + V , where V is the perturbation deforming the operator H 0 . When H = H 0 + V has the continuous spectrum, an effective way of studying its spectral properties is to construct a generalized Fourier tranform associated with H. To accomplish this idea, it is necessary that the Fourier transform for H 0 can be constructed easily. For example, it is the case for the Laplacian −∆ on R n . If the perturbation term V is an operator on the same Hilbert space as for H 0 and is not so strong, one can construct the Fourier transform associated with H 0 + V by using the technique of functional analysis and partial differential equations. This is not so easy for operators on hyperbolic manifolds. Even the construction of the Fourier transform associated with the Laplace-Beltrami operator on the hyperbolic space is no longer a trivial work. To construct the Fourier transform on hyperbolic spaces based on the upper half space model or the ball model, one needs deep knowledge of Bessel functions. Under the action of discrete subgroups, the properties of groups will reflect on the structure of manifolds or the construction of generalized eigenfunctions. 0.3. Spectral and scattering theory on hyperbolic manifolds. In the present note, we deal with the spectral theory and the associated forward and inverse problems for Laplace-Beltrami operators on hyperbolic manifolds. Since we are mainly interested in its spectral properties, Selberg's work [Se56] and its developments are beyond our scope. As an approach to the hyperbolic man...

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Section: Spectral and Scattering Theory On Hyperbolic Manifoldsmentioning

confidence: 99%

Introduction to Spectral Theory and Inverse Problem on Asymptotically Hyperbolic Manifolds

Isozaki¹,

Kurylev²

2014

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“…The uniqueness by a local DN map is well solved (e.g., Belishev [4], Eskin [17], [19], Katchlov, Kurylev and Lassas [26], Kurylev and Lassas [28]). The stability estimates in the case where the DN map is considered on the whole lateral boundary were established in, Stefanov and Uhlmann [39], Sun [42], Bellassoued ans Dos Santos Ferriera [10].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field

Bellassoued

2017

Inverse Problems

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ABSTRACT. In this paper we consider the inverse problem of determining on a compact Riemannian manifold the electric potential or the magnetic field in a Schrödinger equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the magnetic Schrödinger equation. We prove in dimension n ě 2 that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the magnetic field and the electric potential and we establish Hölder-type stability.

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“…Being originally proposed for solving the boundary IP for the multidimensional wave equation, the BC method has been successfully applied to all main types of linear equations of mathematical physics (see the review papers [29,30], monograph [31] and references therein). In this paper we use this method in a one-dimensional situation, applying it to the IP for the wave equation on the semi-axis.…”

Section: The Boundary Control Approach To Ips On the Semi-axismentioning

confidence: 99%

Inverse problem on the semi-axis: local approach

Avdonin

Belinskiy

Matthews³

2011

Tamkang J. Math.

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Abstract. We consider the problem of reconstruction of the potential for the wave equation on the semi-axis. We use the local versions of the Gelfand-Levitan and Krein equations, and the linear version of Simon's approach. For all methods, we reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand-side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples verify that the algorithms can be used to reconstruct an unknown potential accurately. The practicality of each approach is briefly discussed. Accurate data preparation is described and implemented.

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Boundary control in reconstruction of manifolds and metrics (the BC method)

Cited by 201 publications

References 33 publications

Introduction to Spectral Theory and Inverse Problem on Asymptotically Hyperbolic Manifolds

Introduction to Spectral Theory and Inverse Problem on Asymptotically Hyperbolic Manifolds

Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field

Inverse problem on the semi-axis: local approach

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