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

We consider a simple molecular-type quantum system in which the nuclei have one degree of freedom and the electrons have two levels. The Hamiltonian has the form H(ε)=− ε4 2 ∂2 ∂y2 + h(y), 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 ε by optimal truncation of the Rayleigh–Schrödinger series. That is, we construct Eε and Ψε, such that Ψε = O(1) and (H(ε) − Eε)Ψε <Λexp(−Γ/ε2), where Γ >0.  2005 Elsevier Inc. All rights reserved