The Born-Oppenheimer approximation for a 1D 2+1 particle system with zero-range interactions
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914579719651328 |
|---|---|
| author | Cacciapuoti, Claudio Posilicano, Andrea Saberbaghi, Hamidreza |
| author_facet | Cacciapuoti, Claudio Posilicano, Andrea Saberbaghi, Hamidreza |
| contents | 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 $E_{n}(\varepsilon)=-α^{2}+|σ_{n}|α^{2}\varepsilon^{2/3}+O(\varepsilon)$, where $α$ is a negative constant that explicitly relates to the physical parameters and $σ_{n}$ 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 $[-\frac{α^2}{4+\varepsilon^{2}},+\infty)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21457 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Born-Oppenheimer approximation for a 1D 2+1 particle system with zero-range interactions Cacciapuoti, Claudio Posilicano, Andrea Saberbaghi, Hamidreza Mathematical Physics Spectral Theory 81Q10, 81Q15, 81Q20, 70F07, 46N50 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 $E_{n}(\varepsilon)=-α^{2}+|σ_{n}|α^{2}\varepsilon^{2/3}+O(\varepsilon)$, where $α$ is a negative constant that explicitly relates to the physical parameters and $σ_{n}$ 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 $[-\frac{α^2}{4+\varepsilon^{2}},+\infty)$. |
| title | The Born-Oppenheimer approximation for a 1D 2+1 particle system with zero-range interactions |
| topic | Mathematical Physics Spectral Theory 81Q10, 81Q15, 81Q20, 70F07, 46N50 |
| url | https://arxiv.org/abs/2506.21457 |