Resonance widths for the molecular predissociation

We consider a semiclassical \(2\times 2\) matrix Schr\"odinger operator of the form \(P=-h^2\Delta {\bf I}_2 + {\rm diag}(V_1(x), V_2(x)) +hR(x,hD_x)\), where \(V_1, V_2\) are real-analytic, \(V_2\) admits a non degenerate minimum at 0, \(V_1\) is non trapping at energy \(V_2(0)=0\), and \(R(x,hD_x)=(r_{j,k}(x,hD_x))_{1\leq j,k\leq 2}\) is a symmetric off-diagonal \(2\times 2\) 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 \(-\Delta + \la V_2"(0)x,x\ra /2\), 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 \(\rho_1 = V_2(0) + (e_1+r_{2,2}(0,0))h + {\mathcal O}(h^2)\) (as \(h\rightarrow 0_+\)) satisfies, $$ \Im \rho_1 = -h^{n_0+(1-n_\Gamma)/2}f(h,\ln\frac1{h})e^{-2S/h}, $$ where \(f(h,\ln\frac1{h}) \sim \sum_{0\leq m\leq\ell} f_{\ell,m}h^\ell(\ln\frac1{h})^m\) is a symbol with \(f_{0,0}>0\), \(S>0\) is the so-called Agmon distance associated with the degenerate metric \(\max(0, \min(V_1,V_2))dx^2\), between 0 and \(\{V_1\leq 0\}\), and \(n_0\geq 1\), \(n_{\Gamma}\geq 0\) are integers that depend on the geometry.