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Transition of the semiclassical resonance widths across a tangential crossing energy-level

Marouane Assal, Setsuro fujiie, Kenta Higuchi

Abstract

We consider a 1D $2\times 2$ matrix-valued operator \eqref{System0} with two semiclassical Schrödinger operators on the diagonal entries and small interactions on the off-diagonal ones. When the two potentials cross at a turning point with contact order $n$, the corresponding two classical trajectories at the crossing level intersect at one point in the phase space with contact order $2n$. We compute the transfer matrix at this point between the incoming and outgoing microlocal solutions and apply it to the semiclassical distribution of resonances at the energy crossing level. It is described in terms of a generalized Airy function. This result generalizes \cite{FMW1} to the tangential crossing and \cite{AFH1} to the crossing at a turning point.

Transition of the semiclassical resonance widths across a tangential crossing energy-level

Abstract

We consider a 1D matrix-valued operator \eqref{System0} with two semiclassical Schrödinger operators on the diagonal entries and small interactions on the off-diagonal ones. When the two potentials cross at a turning point with contact order , the corresponding two classical trajectories at the crossing level intersect at one point in the phase space with contact order . We compute the transfer matrix at this point between the incoming and outgoing microlocal solutions and apply it to the semiclassical distribution of resonances at the energy crossing level. It is described in terms of a generalized Airy function. This result generalizes \cite{FMW1} to the tangential crossing and \cite{AFH1} to the crossing at a turning point.
Paper Structure (6 sections, 10 theorems, 132 equations, 2 figures)

This paper contains 6 sections, 10 theorems, 132 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Assumptions and results
  3. Proof of the microlocal connection formulae
  4. Asymptotics in a smaller range
  5. Airy functions
  6. Solutions to a scalar Schrödinger equation near a simple turning point

Key Result

Theorem 1

Suppose Assumptions H2 and H1, and let $\mathfrak{B}_h$ and $\mathcal{R}$ be given by DEFFF and DEFR with $\delta_1=Lh^{\frac{2}{2n+1}}$, $\delta_2=Lh$ for an arbitrarily fixed $L>0$. Then, for $h>0$ small there exists a bijective map such that for any $E\in \mathfrak{B}_h$ one has uniformly as $h\to0^+$, with $\epsilon=0$ when $n=1$, $\frac{1}{2}$ when $n=2$, and $\frac{3}{4n+3}$ when $n\ge 3$.

Figures (2)

  • Figure 1: The potentials and the classical trajectories
  • Figure 2: Incoming and outgoing trajectories near $(0,0)$

Theorems & Definitions (23)

  • Theorem 1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2
  • Remark 2.4: Improved asymptotics in a smaller region
  • Remark 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 13 more