Resonance widths in a case of multidimensional phase space tunneling

Abstract

We consider a semiclassical $2\times 2$ matrix Schrödinger operator of the form $P=-h^2\Delta {\mathbf{I}}_2+\mathrm{diag}(x_n-\mu ,\tau V_2(x))+hR(x,h{D}_x)$, where $\mu$ and $\tau$ are two small positive constants, $V_2$ is real-analytic and admits a non degenerate minimum at 0, and $R=(r_{j,k}(x,h{D}_x){)}_{1\le j,k\le 2}$ is a symmetric off-diagonal $2\times 2$ matrix of first-order differential operators with analytic coefficients. Then, denoting by $e_1$ the first eigenvalue of $-Δ+ \la τV_2"(0)x,x\ra /2$, and under some ellipticity condition on $r_{1,2}=r_{2,1}^{\ast }$, we show that, for any $\mu$ sufficiently small, and for $0<\tau \le \tau (\mu )$ with some $\tau (\mu )>0$, the unique resonance $\rho$ of $P$ such that $\rho =\tau V_2(0)+(e_1+r_{2,2}(0,0))h+\mathcal{O}(h^2)$ (as $h\to 0_{+}$) satisfies, $$\Im \rho =-h^{\frac{3}{2}}f(h,\ln \frac{1}{h})e^{-2S/h},$$ 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 $f_{0,0}>0$, 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 $p_1:={\xi }^2-x_n-\mu$ and $p_2:={\xi }^2+\tau V_2(x)$ (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 $p_1$.

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