Resonance widths in a case of multidimensional phase space tunneling

TL;DR

This paper analyzes the resonance widths in a semiclassical 2x2 matrix Schrödinger operator with multidimensional phase space tunneling, deriving an explicit asymptotic formula for the imaginary part of the resonance involving complex action and broken instantons.

Contribution

It provides a novel semiclassical analysis of resonance widths in a multidimensional tunneling setting with matrix operators, including explicit asymptotics and the role of complex instantons.

Findings

Resonance width scales as $h^{3/2} e^{-2S/h}$ with explicit prefactors.

Derived an asymptotic expansion for the resonance's imaginary part.

Connected resonance phenomena to complex bicharacteristics and broken instantons.

Abstract

We consider a semiclassical $2\times 2$ matrix Schr\"odinger operator of the form $P=-h^2\Delta {\bf I}_2 + {\rm diag}(x_n-\mu, \tau V_2(x)) +hR(x,hD_x)$, where $\mu$ and $\tau$ are two small positive constants, $V_2$ is real-analytic and admits a non degenerate minimum at 0, and $R=(r_{j,k}(x,hD_x))_{1\leq j,k\leq 2}$ 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 $-\Delta + \la \tau V_2"(0)x,x\ra /2$, and under some ellipticity condition on $r_{1,2}=r_{2,1}^*$, we show that, for any $\mu$ sufficiently small, and for $0<\tau \leq\tau(\mu)$ 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 $h\rightarrow 0_+$) satisfies, $$ \Im \rho = -h^{\frac32}f(h,\ln\frac1{h})e^{-2S/h}, $$ 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 $p_1:=\xi^2 - x_n-\mu$ 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 $p_1$.