#### Vol. 7, No. 5, 2014

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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

$P=-{h}^{2}\Delta {I}_{2}+diag\left({V}_{1}\left(x\right),{V}_{2}\left(x\right)\right)+hR\left(x,h{D}_{x}\right),$

where ${V}_{1}$, ${V}_{2}$ are real-analytic, ${V}_{2}$ admits a nondegenerate minimum at 0 with ${V}_{2}\left(0\right)=0$, ${V}_{1}$ is nontrapping at energy $0$, and $R\left(x,h{D}_{x}\right)={\left({r}_{j,k}\left(x,h{D}_{x}\right)\right)}_{1\le j,k\le 2}$ is a symmetric $2×2$ matrix of first-order pseudodifferential operators with analytic symbols. We also assume that ${V}_{1}\left(0\right)>0$. Then, denoting by ${e}_{1}$ the first eigenvalue of $-\Delta +〈{V}_{2}^{\prime \prime }\left(0\right)x,x〉∕2$, and under some ellipticity condition on ${r}_{1,2}$ and additional generic geometric assumptions, we show that the unique resonance ${\rho }_{1}$ of $P$ such that ${\rho }_{1}=\left({e}_{1}+{r}_{2,2}\left(0,0\right)\right)h+\mathsc{O}\left({h}^{2}\right)$ (as $h\to {0}_{+}$) satisfies

$\Im {\rho }_{1}=-{h}^{{n}_{0}+\left(1-{n}_{\Gamma }\right)∕2}f\left(h,ln\frac{1}{h}\right){e}^{-2S∕h},$

where $f\left(h,ln\frac{1}{h}\right)\sim {\sum }_{0\le m\le \ell }{f}_{\ell ,m}{h}^{\ell }{\left(ln\frac{1}{h}\right)}^{m}$ is a symbol with ${f}_{0,0}>0$, $S>0$ is the so-called Agmon distance associated with the degenerate metric $max\left(0,min\left({V}_{1},{V}_{2}\right)\right)\phantom{\rule{0.3em}{0ex}}d{x}^{2}$, between 0 and $\left\{{V}_{1}\le 0\right\}$, and ${n}_{0}\ge 1$, ${n}_{\Gamma }\ge 0$ 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