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 (x_n−μ,τV₂(x))+hR(x,hD_x),

where μ and τ are two small positive constants, V₂ is real-analytic and admits a nondegenerate minimum at 0, and R=(r_j,k(x,hD_x))_1≤j,k≤2 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+〈τV₂″(0)x,x〉/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≤τ(μ) with some τ(μ)>0, the unique resonance ρ of P such that ρ=τV₂(0)+(e₁+r_2,2(0,0))h+𝒪(h²) (as h→0₊) satisfies

Im ρ=−h^3/2f(h,ln 1/h)e^−2S/h,

where f(h,ln 1/h)~Σ_0≤m≤ℓf_ℓ,mh^ℓ(ln 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 p₁:=ξ²−x_n−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 p₁⁻¹(0) of p₁.