Semiclassical asymptotic distribution of resonances created by homoclinic and heteroclinic trajectories

Abstract

Here we review a part of the joint work with Jean-François Bony, Thierry Ramond and Maher Zerzeri [Astérisque 405 (2018), pp. vii+314] on the semiclassical distribution of resonances for Schrödinger operators in a complex neighborhood of a fixed energy, assuming that the trapped set of the underlying classical dynamics on the energy surface consists of hyperbolic fixed points and associated homoclinic and heteroclinic trajectories. Based on a microlocal theory of solutions near a hyperbolic fixed point established by Bony, Fujiié, Ramond, and Zerzeri [J. Funct. Anal. 252 (2007), pp. 68–125], we compute the quantization condition of resonances and prove its accuracy. We will see in particular that the principal term of the imaginary part of resonances is − D 0 h -D_0 h , where D 0 D_0 is the minimum over all ‘cycles’ in the trapped set of what we call damping index defined in terms of the exponents at the hyperbolic fixed points therein. We will also see the subprincipal term of order h / | log ⁡ h | h/|\log h| . In this scale, resonances present various interesting distributions reflecting the geometry of the classical dynamics.