Resonance widths for the molecular predissociation

Abstract

We consider a semiclassical $2\times 2$ matrix Schrödinger operator of the form $P=-h^2\Delta {\mathbf{I}}_2+\mathrm{diag}(V_1(x),V_2(x))+hR(x,h{D}_x)$, 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 $R(x,h{D}_x)=(r_{j,k}(x,h{D}_x){)}_{1\le j,k\le 2}$ 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 $-Δ+ \la V_2"(0)x,x\ra /2$, 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 $h\to 0_{+}$) satisfies, $$\Im {\rho }_1=-h^{n_0+(1-n_{\Gamma })/2}f(h,\ln \frac{1}{h})e^{-2S/h},$$ where $f(h,\ln \frac{1}{h})\sim {\sum }_{0\le m\le \ell }{f}_{\ell ,m}{h}^{\ell }(\ln \frac{1}{h}{)}^m$ is a symbol with $f_{0,0}>0$, $S>0$ is the so-called Agmon distance associated with the degenerate metric $\max (0,\min (V_1,V_2))d{x}^2$, between 0 and $\{V_1\le 0\}$, and $n_0\ge 1$, $n_{\Gamma }\ge 0$ are integers that depend on the geometry.

37 pages, no figure

Cited by in corpus (3)

• Molecular predissociation resonances below an energy level crossing
• Energy-time uncertainty principle and lower bounds on sojourn time
• Scattering theory for multistate Schrödinger operators