Resonance widths for the molecular predissociation

where $V_{1}$ , $V_{2}$ are real-analytic, $V_{2}$ admits a nondegenerate minimum at 0 with $V_{2} (0) = 0$ , $V_{1}$ is nontrapping at energy $0$ , and $R (x, h D_{x}) = {(r_{j, k} (x, h D_{x}))}_{1 \leq j, k \leq 2}$ is a symmetric $2 \times 2$ matrix of first-order pseudodifferential operators with analytic symbols. We also assume that $V_{1} (0) > 0$ . Then, denoting by $e_{1}$ the first eigenvalue of $- Δ + 〈 V_{2}^{''} (0) x, x 〉 ∕ 2$ , and under some ellipticity condition on $r_{1, 2}$ and additional generic geometric assumptions, we show that the unique resonance $ρ_{1}$ of $P$ such that $ρ_{1} = (e_{1} + r_{2, 2} (0, 0)) h + O (h^{2})$ (as $h \to 0_{+}$ ) satisfies

ℑ ρ_{1} = - h^{n_{0} + (1 - n_{Γ}) ∕ 2} f (h, ln \frac{1}{h}) e^{- 2 S ∕ h},

where $f (h, ln \frac{1}{h}) \sim \sum_{0 \leq m \leq ℓ} f_{ℓ, m} h^{ℓ} {(ln \frac{1}{h})}^{m}$ is a symbol with $f_{0, 0} > 0$ , $S > 0$ is the so-called Agmon distance associated with the degenerate metric $max (0, min (V_{1}, V_{2})) d x^{2}$ , between 0 and ${V_{1} \leq 0}$ , and $n_{0} \geq 1$ , $n_{Γ} \geq 0$ are integers that depend on the geometry.

Keywords

resonances, Born–Oppenheimer approximation, eigenvalue crossing, microlocal analysis

Mathematical Subject Classification 2010

Primary: 35P15, 35C20

Secondary: 35S99, 47A75

Milestones

Received: 28 October 2011

Revised: 23 May 2014

Accepted: 30 June 2014

Published: 27 September 2014

Authors

Alain Grigis
Departement de Mathematiques Universite Paris 13 - Institut Galilee Avenue Jean-Baptiste Clement 93430 Villetaneuse France

André Martinez
Dipartimento di Matematica Università di Bologna Piazza di Porta San Donato I-40127 Bologna Italy

Vol. 7, No. 5, 2014

Alain Grigis and André Martinez

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors