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 , where is the dimension of the manifold.
We consider perturbations of a stratified medium , where the operator studied is . The function is a perturbation of , which is constant for sufficiently large and satisfies some other conditions. Under certain restrictions on the perturbation , 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 from knowledge of and the singularities of the scattering matrix at fixed energy....
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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