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Here we define and prove some properties of the semi-classical wavefront set. We also define and study semi-classical Fourier integral operators and prove a generalization of Egorov's Theorem to manifolds of different dimensions.

We consider scattering by general compactly supported semi-classical perturbations of the Euclidean Laplace-Beltrami operator. We show that if the suitably cutoff resolvent of the Hamiltonian quantizes a Lagrangian relation on the product cotangent bundle, the scattering amplitude quantizes the natural scattering relation. In the case when the resolvent is… (More)

We consider scattering by short range perturbations of the semi-classical Laplacian. We prove that when a polynomial bound on the resolvent holds the scattering amplitude is a semi-classical Fourier integral operator associated to the scattering relation near a non-trapped ray. Compared to previous work, we allow the scattering relation to have more general… (More)

We study the scattering amplitude for Schrödinger operators at a critical energy level, which is a unique non-degenerate maximum of the potential. We do not assume that the maximum point is non-resonant and use results by Bony, Fujiié, Ramond and Zerzeri to analyze the contributions of the trapped trajectories. We prove a semiclassical expansion of the… (More)

The existence of the Laplace-Beltrami operator has allowed mathematicians to carry out Fourier analysis on Riemannian manifolds [2]. We recall that the Laplace-Beltrami operator ∆ on a compact Riemannian manifold has a discrete set of eigenvalues {λ j } ∞ j=1 , which satisfies λ j → ∞ as j → ∞. This is known as the spectrum of the Laplace-Beltrami operator.… (More)

We compute the scattering amplitude for Schrödinger operators at a critical energy level, corresponding to the maximum point of the potential. We follow [30], using Isozaki-Kitada's representation formula for the scattering amplitude, together with results from [5] in order to analyze the contribution of trapped trajectories.

We study the semi-classical behavior of the spectral function of the Schrödinger operator with short range potential. We prove that the spectral function is a semi-classical Fourier integral operator quantizing the forward and backward Hamiltonian flow relations of the system. Under a certain geometric condition we explicitly compute the phase in an… (More)

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