Exponentially accurate semiclassical asymptotics of low-lying eigenvalues for 2×2 matrix Schrödinger operators

where h(y) is a 2×2 real symmetric matrix. Near a local minimum of an electron level E(y) that is not at a level crossing, we construct quasimodes that are exponentially accurate in the square of the Born-Oppenheimer parameter ɛ by optimal truncation of the Rayleigh-Schrödinger series. That is, we construct Eɛ and Ψɛ, such that ‖Ψɛ‖=O(1) and ‖(H(ɛ)−Eɛ)Ψɛ‖<Λexp(−Γ/ɛ2), where Γ>0.