Vol. 7, No. 5, 2014

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ISSN: 1948-206X (e-only)
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Resonance widths for the molecular predissociation

Alain Grigis and André Martinez

Vol. 7 (2014), No. 5, 1027–1055
Abstract

We consider a semiclassical 2 × 2 matrix Schrödinger operator of the form

P = h2ΔI 2 + diag(V 1(x),V 2(x)) + hR(x,hDx),

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,hDx) = (rj,k(x,hDx))1j,k2 is a symmetric 2 × 2 matrix of first-order pseudodifferential operators with analytic symbols. We also assume that V 1(0) > 0. Then, denoting by e1 the first eigenvalue of Δ + V 2(0)x,x2, and under some ellipticity condition on r1,2 and additional generic geometric assumptions, we show that the unique resonance ρ1 of P such that ρ1 = (e1 + r2,2(0,0))h + O(h2) (as h 0+) satisfies

ρ1 = hn0+(1nΓ)2f(h,ln 1 h)e2Sh,

where f(h,ln 1 h) 0mf,mh(ln 1 h)m is a symbol with f0,0 > 0, S > 0 is the so-called Agmon distance associated with the degenerate metric max(0,min(V 1,V 2))dx2, between 0 and {V 1 0}, and n0 1, nΓ 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