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 1, where is proportional to the square root of the mass ratio. We show that the n-th eigenvalue behaves as E₍ () =-α^2+|σ₍|\, α^2^2/3+O () \, , where α is a negative constant that explicitly relates to the physical parameters and σ₍ 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+^{2}\, , +).