Molecular predissociation resonances below an energy level crossing
We study the resonances of $2\times 2$ systems of one dimensional Schrodinger operators which are related to the mathematical theory of molecular predissociation. We determine the precise positions of the resonances with real parts below the energy where bonding and anti-bonding potentials intersect transversally. In particular, we find that imaginary parts (widths) of the resonances are exponentially small and that the indices are determined by Agmon distances for the minimum of two potentials.
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.
Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
The aim of the present paper is to bridge the gap between the Bakry–Emery and the Lott–Sturm–Villani approaches to provide synthetic and abstract notions of lower Ricci curvature bounds. 
 
We start from a strongly local Dirichlet form E admitting a Carre du champ Γ in a Polish measure space (X,m) and a canonical distance dE that induces the original topology of X. We first characterize the distinguished class of Riemannian Energy measure spaces, where E coincides with the Cheeger energy induced by dE and where every function f with Γ(f)≤1 admits a continuous representative. 
 
In such a class, we show that if E satisfies a suitable weak form of the Bakry–Emery curvature dimension condition BE(K,∞) then the metric measure space (X,d,m) satisfies the Riemannian Ricci curvature bound RCD(K,∞) according to [Duke Math. J. 163 (2014) 1405–1490], thus showing the equivalence of the two notions. 
 
Two applications are then proved: the tensorization property for Riemannian Energy spaces satisfying the Bakry–Emery BE(K,N) condition (and thus the corresponding one for RCD(K,∞) spaces without assuming nonbranching) and the stability of BE(K,N) with respect to Sturm–Gromov–Hausdorff convergence.