The Born–Oppenheimer approximation for a 1D 2+1 particle system with zero-range interactions

We study the self-adjoint Hamiltonian that models the quantum dynamics of a one-dimensional three-body system consisting of a light particle interacting with two heavy ones through a zero-range force. For an attractive interaction we determine the behavior of the eigenvalues below the essential spectrum in the regime ɛ ≪ 1, where ɛ is proportional to the square root of the mass ratio. We show that the n-th eigenvalue behaves as En(ɛ) = −α2 + |σn| α2ɛ2/3 + O(ɛ), where α is a negative constant that explicitly relates to the physical parameters and σn is either the n-th extremum or the n-th zero of the Airy function Ai, depending on the kind (respectively, bosons or fermions) of the two heavy particles. Additionally, we prove that the essential spectrum coincides with the half-line [−α24+ε2,+∞).