Semiclassical scattering amplitude at the maximum of the potential

Article type: Research Article

Authors: Alexandrova, Ivana | Bony, Jean-François | Ramond, Thierry

Affiliations: Department of Mathematics, East Carolina University, Greenville, NC 27858, USA. E-mail: alexandrovai@ecu.edu | Institut de Mathématiques de Bordeaux (UMR CNRS 5251), Université de Bordeaux I, 33405 Talence, France. E-mail: bony@math.u-bordeaux1.fr | Mathématiques, Université Paris Sud (UMR CNRS 8628), 91405 Orsay, France. E-mail: thierry.ramond@math.u-psud.fr

Abstract: 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 scattering amplitude and compute its leading term. We show that it has different orders of magnitude in specific regions of phase space. We also prove upper and lower bounds for the resolvent in this setting.

Keywords: scattering amplitude, critical energy, Schrödinger equation

DOI: 10.3233/ASY-2008-0877

Journal: Asymptotic Analysis, vol. 58, no. 1-2, pp. 57-125, 2008