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arXiv · 2506.21457

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

Claudio Cacciapuoti·Andrea Posilicano·Hamidreza Saberbaghi
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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 $[-\frac{\alpha^2}{4+\varepsilon^{2}},+\infty)$.

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Claudio Cacciapuoti, Andrea Posilicano, Hamidreza Saberbaghi. 2025-06-26. The Born-Oppenheimer approximation for a 1D 2+1 particle system with zero-range interactions. https://doi.org/10.1063/5.0287840

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