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Resonances for semiclassical matrix Schrödinger operators created by non-trapping trajectories

樋口 健太 立命館大学 DOI:info:doi/10.34382/00016912

2022.05.12

概要

Quantum resonances of a Schrödinger operator are defined to be the poles of the resolvent. Each resonance is associated with a quasi-bound state, and its imaginary part (called width) describes the reciprocal of the half lifetime of the state. In the semiclassical limit, where the Plank constant ℏ tends to 0 as a small parameter, it is expected to recover the underlying classical mechanics (Bohr's correspondence principle). The study of the asymptotic distribution of resonances has been developed since 1980's in relation with the underlying classical dynamics. In particular, the non-existence of resonances near a non- trapping real energy was proved in 1987 by Helffer and Sjöstrand for analytic potentials (a real energy is said to be non-trapping when any classical trajectory on the energy surface goes to infinity as time goes to plus and minus infinity). This theorem was extended to smooth potentials by width of order ℏ log(1/ℏ).

The author found that this is not the case in general for matrix Schrödinger operators. He studied a one dimensional 2 × 2 Born-Oppenheimer approximation model, where the diagonal entries consist of two Schrödinger operators while the anti-diagonals are small interactions. He showed the existence and the precise asymptotic distribution of resonances with width of order ℏ log(1/ℏ) near an energy non- trapping for both Schrödinger operators. These resonances are created by a periodic trajectory composed by two non-trapping classical trajectories in the phase space. Such a phenomenon is due to a so-called energy-level crossing, which is a typical and important subject in quantum chemistry for the study of　molecular predissociation and non-adiabatic transition.

The aim of this thesis is to present two approaches to this problem. One is a classical approach, and the other is a microlocal approach. The classical approach consists of the construction of local solutions on both side of the crossing point and compute the Wronskian of these solutions. On the other hand, the microlocal approach reduces the problem to the study of the transfer matrix at a crossing point in the phase space which permits us to know the microlocal behavior of solutions on the outgoing trajectories in terms of that on the incoming trajectories. In addition to the results in his previous work, we supply this thesis with further developments. The general case where two classical trajectories make a graph structure with finitely many crossings of finite order degeneracy is discussed.

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