Resonance widths in a case of multidimensional phase space tunneling

Motivated by the study of resonances for molecular systems in the Born–Oppenheimer approximation, we consider a semiclassical 2 × 2 matrix Schrödinger operator of the form P = −h2ΔI2 + diag(xn − μ, τV2(x)) + hR(x,hDx), where μ and τ are two small positive constants, V2 is real-analytic and admits a nondegenerate minimum at 0, and R = (rj,k(x,hDx))1j,k2 is a symmetric off-diagonal 2 × 2 matrix of first-order differential operators with analytic coefficients. Then, denoting by e1 the first eigenvalue of −Δ + τV 2 (0)x, x/2, and under some ellipticity condition on r1,2 = r ∗ 2,1, we show that, for any μ sufficiently small, and for 0 < τ τ (μ) with some τ (μ) > 0, the unique resonance ρ of P such that ρ = τV2(0) + (e1 + r2,2(0, 0))h + O(h2) (as h → 0+) satisfies Im ρ = −h 32 f h, ln 1 h e−2S/h, where f(h, ln 1 h ) ∼ 0m f,mh(ln 1 h )m is a symbol with f0,0 > 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 p1 := ξ2 − xn − μ and p2 := ξ2 + τV2(x) (broken instanton). This broken instanton is described in terms of the outgoing and incoming complex Lagrangian manifolds associated with p2 at the point (0, 0), and their intersections with the characteristic set p −1 1 (0) of p1.