Semiclassical Resonance Asymptotics for Systems With Degenerate Crossings of Classical Trajectories

Abstract

Abstract This paper is concerned with the asymptotics of resonances in the semiclassical limit $h\to 0^{+}$ for two-by-two matrix Schrödinger operators in one dimension. We study the case where the two underlying classical Hamiltonian trajectories cross tangentially in the phase space. In the setting that one of the classical trajectories is a simple closed curve whereas the other one is non-trapping, we show that the imaginary part of the resonances is of order $h^{(m_{0}+3)/(m_{0}+1)}$, where $m_{0}$ is the maximal contact order of the crossings. This principal order comes from the subprincipal term of the transfer matrix at crossing points, which describes the propagation of microlocal solutions from one trajectory to the other. In addition, we compute explicitly the leading coefficient of the resonance widths.