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Semiclassical resonances for matrix Schrödinger operators with vanishing interactions at crossings of classical trajectories

Vincent Louatron

Abstract

We study the semiclassical distribution of resonances of a $2 \times 2$ matrix Schrödinger operator, obtained as a reduction of an Hamiltonian when studying molecular predissociation models under the Born-Oppenheimer approximation. The energy considered is above the energy-level crossing of the two associated classical trajectories, and is respectively trapping and non-trapping for those trajectories. Under a condition between the contact order $m$ of the crossings and the vanishing order $k$ of the interaction term at the crossing points, we show that, asymptotically in the semiclassical limit $h \to 0^+$, the imaginary part of the resonances is of size $h^{1+2(k+1)/(m+1)}$ in the general case and shrinks to $h^{1+2(k+2)/(m+1)}$ when both $k$ and $m$ are odd. We also compute the first term of the associated asymptotic expansions.

Semiclassical resonances for matrix Schrödinger operators with vanishing interactions at crossings of classical trajectories

Abstract

We study the semiclassical distribution of resonances of a matrix Schrödinger operator, obtained as a reduction of an Hamiltonian when studying molecular predissociation models under the Born-Oppenheimer approximation. The energy considered is above the energy-level crossing of the two associated classical trajectories, and is respectively trapping and non-trapping for those trajectories. Under a condition between the contact order of the crossings and the vanishing order of the interaction term at the crossing points, we show that, asymptotically in the semiclassical limit , the imaginary part of the resonances is of size in the general case and shrinks to when both and are odd. We also compute the first term of the associated asymptotic expansions.
Paper Structure (9 sections, 10 theorems, 106 equations, 5 figures)

This paper contains 9 sections, 10 theorems, 106 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Framework and main results
  3. Microlocal connection formula at a crossing point
  4. Definition of the transfer matrix
  5. Microlocal connection formula at a crossing point
  6. Proof of the main results
  7. Crossing at a turning point
  8. Generalization to an interaction $U=r_0(x) + ihr_1(x)D_x$
  9. A degenerate stationary phase estimate

Key Result

Theorem 2.4

Assume that assumptions Assumption:AnalyticContinuation up to Assumption:FiniteOrderInteraction hold true and that either Case:Ia or Case:Ib holds true. Then for all small $h >0$, there exists a bijective map $z_h: \mathfrak{B}_h \to \mathop{\mathrm{Res}}\nolimits_h(P)$ such that for any $E \in \mat and with $s := \min\left( 1/3, 1/(m+1)\right)$ and Here $\mathcal{S} = \mathcal{S}(E) := 2\left(\

Figures (5)

  • Figure 1: Potential crossing at $x=0$ - case \ref{['Case:Ia']}
  • Figure 2: Phase space crossings of the associated classical trajectories case \ref{['Case:Ia']}
  • Figure 3: Labeling around the upper crossing point $\rho_+$
  • Figure 4: Potential crossing - case \ref{['Case:II']}
  • Figure 5: Associated phase space crossings - case \ref{['Case:II']}

Theorems & Definitions (25)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 15 more