Semiclassical resonance asymptotics for systems with degenerate crossings of classical trajectories

TL;DR

This paper analyzes the asymptotic behavior of resonances in semiclassical matrix Schrödinger operators with degenerate crossing trajectories, revealing how the resonance widths depend on contact order and classical trajectory properties.

Contribution

It provides a new asymptotic formula for resonance widths in systems with tangential trajectory crossings, incorporating subprincipal transfer matrix terms and generalized classical trajectories.

Findings

Resonance imaginary parts scale as h^{(m_0+3)/(m_0+1)}.

Explicit calculation of leading coefficients of resonance widths.

Dependence of resonance behavior on contact order and classical trajectory structure.

Abstract

This paper is concerned with the asymptotics of resonances in the semiclassical limit $h\to 0^+$ for $2\times 2$ matrix Schr\"odinger 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 terms of the transfer matrix at crossing points which describe the propagation of microlocal solutions from one trajectory to the other. In addition, we compute explicitly the leading coefficient of the resonance widths in terms of the probability amplitudes associated with all the \textit{generalized classical trajectories} escaping to infinity from the closed trajectory.