Displaying similar documents to “Recovering the total singularity of a conormal potential from backscattering data”

Composition of some singular Fourier integral operators and estimates for restricted X -ray transforms

Allan Greenleaf, Gunther Uhlmann (1990)

Annales de l'institut Fourier

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We establish a composition calculus for Fourier integral operators associated with a class of smooth canonical relations C ⊂ ( T * X ∖ 0 ) × ( T * Y ∖ 0 ) . These canonical relations, which arise naturally in integral geometry, are such that π : C → T * Y is a Whitney fold and ρ : C → T * X is a blow-down mapping. If A ∈ I m ( C ) , B ∈ I m ' ( C t ) , then B A ∈ I m + m ' , 0 ( Δ , Λ ) a class of pseudodifferential operators with singular symbols. From this follows L 2 boundedness of A with a loss of 1/4 derivative.

The resolvent for Laplace-type operators on asymptotically conic spaces

Andrew Hassell, András Vasy (2001)

Annales de l’institut Fourier

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Let X be a compact manifold with boundary, and g a scattering metric on X , which may be either of short range or “gravitational” long range type. Thus, g gives X the geometric structure of a complete manifold with an asymptotically conic end. Let H be an operator of the form H = Δ + P , where Δ is the Laplacian with respect to g and P is a self-adjoint first order scattering differential operator with coefficients vanishing at ∂ X and satisfying a “gravitational” condition. We define a symbol calculus...

Propagation of singularities for operators with multiple involutive characteristics

Johannes Sjöstrand (1976)

Annales de l'institut Fourier

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Let P be a classical pseudodifferential operator of order m on a paracompact C ∞ manifold X . Let p m be the principal symbol and assume that Σ = p m - 1 ( 0 ) is an involutive C ∞ sub-manifold of T * X ∖ 0 , satisfying a certain transversality condition. We assume that p m vanishes exactly to order M on Σ and that the derivatives of order M satisfy a certain condition, inspired from the Calderòn uniqueness theorem (usually empty when M = 2 ). Suppose that a Levi condition is valid for the lower order symbols. If u ∈ 𝒟 ' ( X ) , P u ∈ C ∞ ( X ) , then...

Propagation of singularities in many-body scattering in the presence of bound states

András Vasy (1999)

Journées équations aux dérivées partielles

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In these lecture notes we describe the propagation of singularities of tempered distributional solutions u ∈ 𝒮 ' of ( H - λ ) u = 0 , where H is a many-body hamiltonian H = Δ + V , Δ ≥ 0 , V = ∑ a V a , and λ is not a threshold of H , under the assumption that the inter-particle (e.g. two-body) interactions V a are real-valued polyhomogeneous symbols of order - 1 (e.g. Coulomb-type with the singularity at the origin removed). Here the term “singularity” provides a microlocal description of the lack of decay at infinity. Our result is...

Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I)

Maurice De Gosson (1982)

Annales de l'institut Fourier

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This work is devoted to a systematic study of the microlocal regularity properties of pseudo-differential operators with the transmission property. We introduce a “boundary singular spectrum”, denoted ∂ W F ω ( u ) for distributions u ∈ D ' ( R + n ) , regular in the normal variable x n (thus, ∂ W F ω ( u ) = ∅ means that u ∈ ∩ s + t = 1 / 2 H s + t near the boundary), and it is shown that ∂ W F ω - m [ P ( u 0 ) x n > 0 ] is a subset of ∂ W F ( u ) if P has degree m and the transmission property. We finally prove that these results can bef used to examinate the (microlocal) regularity of the solutions...