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Spectral asymptotics for manifolds with cylindrical ends

Tanya ChristiansenMaciej Zworski — 1995

Annales de l'institut Fourier

The spectrum of the Laplacian on manifolds with cylindrical ends consists of continuous spectrum of locally finite multiplicity and embedded eigenvalues. We prove a Weyl-type asymptotic formula for the sum of the number of embedded eigenvalues and the scattering phase. In particular, we obtain the optimal upper bound on the number of embedded eigenvalues less than or equal to r 2 , 𝒪 ( r n ) , where n is the dimension of the manifold.

Recovering Asymptotics at Infinity of Perturbations of Stratified Media

Tanya ChristiansenMark S. Joshi — 2000

Journées équations aux dérivées partielles

We consider perturbations of a stratified medium x n - 1 × y , where the operator studied is c 2 ( x , y ) Δ . The function c is a perturbation of c 0 ( y ) , which is constant for sufficiently large | y | and satisfies some other conditions. Under certain restrictions on the perturbation c , we give results on the Fourier integral operator structure of the scattering matrix. Moreover, we show that we can recover the asymptotic expansion at infinity of c from knowledge of c 0 and the singularities of the scattering matrix at fixed energy....

Scattering on stratified media: the microlocal properties of the scattering matrix and recovering asymptotics of perturbations

Tanya ChristiansenM. S. Joshi — 2003

Annales de l’institut Fourier

The scattering matrix is defined on a perturbed stratified medium. For a class of perturbations, its main part at fixed energy is a Fourier integral operator on the sphere at infinity. Proving this is facilitated by developing a refined limiting absorption principle. The symbol of the scattering matrix determines the asymptotics of a large class of perturbations.

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