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

TL;DR

This paper analyzes the spectral properties of a 1D three-body quantum system with zero-range interactions, deriving eigenvalue behaviors and the essential spectrum in the small mass ratio regime.

Contribution

It provides explicit asymptotic formulas for eigenvalues and characterizes the essential spectrum for a 1D three-body system with zero-range forces, considering different particle statistics.

Findings

Eigenvalues below the essential spectrum behave as $E_{n}( ext{varepsilon})=- ext{alpha}^2 + | ext{sigma}_n| ext{alpha}^2 ext{varepsilon}^{2/3} + O( ext{varepsilon})$.

The essential spectrum is the half-line $[-rac{ ext{alpha}^2}{4+ ext{varepsilon}^2},+ ablafty)$.

Eigenvalues depend on the zeros or extrema of the Airy function Ai, depending on particle statistics.

Abstract

We study the self-adjoint Hamiltonian that models the quantum dynamics of a one-dimensional (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 $\varepsilon\ll 1$, where $\varepsilon$ is proportional to the square root of the mass ratio. We show that the $n$-th eigenvalue behaves as $E_{n}(\varepsilon)=-\alpha^{2}+|\sigma_{n}|\alpha^{2}\varepsilon^{2/3}+O(\varepsilon)$, where $\alpha$ is a negative constant that explicitly relates to the physical parameters and $\sigma_{n}$ 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 $[-\frac{\alpha^2}{4+\varepsilon^{2}},+\infty)$.