Forward and inverse scattering on manifolds with asymptotically cylindrical ends

Isozaki, Hiroshi; Kurylev, Yaroslav; Lassas, Matti

doi:10.1016/j.jfa.2009.11.009

Cited by 29 publications

(28 citation statements)

References 65 publications

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“…144, 115]. By considering a subsequence we may assume that z j ∈ Γ since z j → y 0 ∈ Γ. Lemma 8 and the inequality (26) imply that…”

Section: 4mentioning

confidence: 99%

Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets

Lassas¹,

Oksanen²

2014

Duke Math. J.

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We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary (M, g) from a restriction Λ S,R of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here S and R are open sets in ∂M and the restriction Λ S,R corresponds to the case where the Dirichlet data is supported on R + × S and the Neumann data is measured on R + × R. In the novel case where S ∩ R = ∅, we show that Λ S,R determines the manifold (M, g) uniquely, assuming that the wave equation is exactly controllable from the set of sources S. Moreover, we show that the exact controllability can be replaced by the Hassell-Tao condition for eigenvalues and eigenfunctions, that is,

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“…144, 115]. By considering a subsequence we may assume that z j ∈ Γ since z j → y 0 ∈ Γ. Lemma 8 and the inequality (26) imply that…”

Section: 4mentioning

confidence: 99%

Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets

Lassas¹,

Oksanen²

2014

Duke Math. J.

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“…2, §5, §6 and Appendix A. The inverse scattering from asymptotically (Euclidean) cylindrical ends has been studied in [IKL10]. In practical situation, this problem includes that of wave guides.…”

Section: Appendix a Radon Transform And Propagation Of Singularities mentioning

confidence: 99%

“…The BC-method was first applied to compact manifolds ([BeKu92]), and was extended to non-compact manifolds (see e.g. [KKL04], [IKL10]). …”

Section: This Implies Fmentioning

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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“…The crucial idea for the inverse problem part is the boundary control method. Just like our previous paper for the inverse scattering on manifolds with cylindrical ends [25], we reduce the issue to the inverse boundary value problem from an artificial boundary in the end M 1 . The new ingredient in this paper is the argument around conic singularities based on the explicit form of the metric (1.2).…”

Section: 2mentioning

confidence: 99%

Conic singularities, generalized scattering matrix, and inverse scattering on asymptotically hyperbolic surfaces

Isozaki

Kurylev

Lassas

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface M = Γ\H 2 associated with a Fuchsian group of the 1st kind Γ containing parabolic elements. M is then non-compact, and has a finite number of cusps and elliptic singular points, which is regarded as a hyperbolic orbifold. We introduce a class of Riemannian surfaces with conical singularities on its finite part, having cusps and regular ends at infinity, whose metric is asymptotically hyperbolic. By observing solutions of the Helmholtz equation at the cusp, we define a generalized S-matrix. We then show that this generalized S-matrix determines the Riemannian metric and the structure of conical singularities.

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Forward and inverse scattering on manifolds with asymptotically cylindrical ends

Cited by 29 publications

References 65 publications

Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets

Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets

Introduction to Spectral Theory and Inverse Problem on Asymptotically Hyperbolic Manifolds

Conic singularities, generalized scattering matrix, and inverse scattering on asymptotically hyperbolic surfaces

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