Vol. 7, No. 5, 2014

Download this article
Download this article For screen
For printing
Recent Issues

Volume 12
Issue 8, 1891–2146
Issue 7, 1643–1890
Issue 7, 1397–1644
Issue 6, 1397–1642
Issue 5, 1149–1396
Issue 4, 867–1148
Issue 3, 605–866
Issue 2, 259–604
Issue 1, 1–258

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Subscriptions
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
 
Other MSP Journals
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