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Resonance widths in a case of multidimensional phase space tunneling

We consider a semiclassical $2\times 2$ matrix Schrödinger operator of the form $P=-h^2Δ{\bf I}_2 + {\rm diag}(x_n-μ, τV_2(x)) +hR(x,hD_x)$, where $μ$ and $τ$ are two small positive constants, $V_2$ is real-analytic and admits a non degenerate minimum at 0, and $R=(r_{j,k}(x,hD_x))_{1\leq j,k\leq 2}$ is a symmetric off-diagonal $2\times 2$ matrix of first-order differential operators with analytic coefficients. Then, denoting by $e_1$ the first eigenvalue of $-Δ+ \la τV_2&#34;(0)x,x\ra /2$, and under some ellipticity condition on $r_{1,2}=r_{2,1}^*$, we show that, for any $μ$ sufficiently small, and for $0<τ\leqτ(μ)$ with some $τ(μ)>0$, the unique resonance $ρ$ of $P$ such that $ρ= τV_2(0) + (e_1+r_{2,2}(0,0))h + {\mathcal O}(h^2)$ (as $h\rightarrow 0_+$) satisfies, $$ \Im ρ= -h^{\frac32}f(h,\ln\frac1{h})e^{-2S/h}, $$ 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 $f_{0,0}>0$, and $S$ is the imaginary part of the complex action along some convenient closed path containing $(0,0)$ and consisting of a union of complex nul-bicharacteristics of $p_1:=ξ^2 - x_n-μ$ and $p_2:=ξ^2 +τV_2(x)$ (broken instanton). This broken instanton is described in terms of the outgoing and incoming complex Lagrangian manifolds associated with $p_2$ at the point $(0,0)$, and their intersections with the characteristic set $p_1^{-1}(0)$ of $p_1$.