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 1D 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 ε 1, where ε is proportional to the square root of the mass ratio. We show that the n-th eigenvalue behaves as Eₙ(ε)=-α²+|σₙ|\,α²ε2/3+O(ε)\,, 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\,,+∞).