Resonance widths for the molecular predissociation

We consider a semiclassical 2 x 2 matrix Schrodinger operator of the form P = -h(2)Delta I-2 + diag(V-1(x), V-2(x)) + hR(x, hD(x)), where V-1, V-2 are real-analytic, V-2 admits a nondegenerate minimum at 0 with V-2(0) = 0, V-1 is nontrapping at energy 0, and R(x, hD(x)) = (r (j,k)(x, hD(x)))(1 <= j,k <= 2) is a symmetric 2 x 2 matrix of first-order pseudodifferential operators with analytic symbols. We also assume that V-1(0) > 0. Then, denoting by e(1) the first eigenvalue of -Delta + (V-2(")(0) x, x >/2, and under some ellipticity condition on r(1,2) and additional generic geometric assumptions, we show that the unique resonance rho(1) of P such that rho(1) = (e(1) + r(2,2)(0, 0))h+O(h(2)) (as h -> 0(+)) satisfies Im rho(1) = -h(n0+(1-n Gamma)/2) f(h, ln 1/h) e(-2S/h), where f(h, ln 1/h) similar to Sigma(0 <= m <= l) fl,mh(l) (ln 1/h)(m) is a symbol with f(0, 0) > 0, S > 0 is the so-called Agmon distance associated with the degenerate metric max(0, min(V-1, V-2)) dx(2), between 0 and {V-1 <= 0}, and n(0) >= 1, n(Gamma) >= 0 are integers that depend on the geometry.