Geodesic
Resolvent and Scattering Matrix at the Maximum of the Potential
Serdica Mathematical Journal, Tome 34 (2008) no. 1, pp. 267-310

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We study the microlocal structure of the resolvent of the semiclassical Schrödinger operator with short range potential at an energy which is a unique non-degenerate global maximum of the potential. We prove that it is a semiclassical Fourier integral operator quantizing the incoming and outgoing Lagrangian submanifolds associated to the fixed hyperbolic point. We then discuss two applications of this result to describing the structure of the spectral function and the scattering matrix of the Schrödinger operator at the critical energy.
Keywords: Scattering Matrix, Resolvent, Spectral Function, Schrödinger Equation, Fourier Integral Operator, Critical Energy 
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Alexandrova, Ivana; Bony, Jean-François; Ramond, Thierry. Resolvent and Scattering Matrix at the Maximum of the Potential. Serdica Mathematical Journal, Tome 34 (2008) no. 1, pp. 267-310. http://geodesic.mathdoc.fr/item/SMJ2_2008_34_1_a10/