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 x 2 matrix Schrodinger operator of the form P = -h(2)Delta I-2 + 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 nondegenerate minimum at 0, and R = (r(j,k)(x, hD(x)))(1 <= j,k <= 2) is a symmetric off-diagonal 2 x 2 matrix of first-order differential operators with analytic coefficients. Then, denoting by e(1) the first eigenvalue of -Delta + /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 <= tau(mu) with some tau(mu) > 0, the unique resonance rho of P such that rho = tau V-2(0) vertical bar (e(1) vertical bar r(2,2)(0, 0))h vertical bar O(h(2)) (as h -> 0(+)) satisfies Im rho = -h(3/2) f(h, ln 1/h)e(-2S/h), where f(h, ln 1/h) similar to Sigma(0 <= m <= l) f(l,m)h(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 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).