Rigorous derivation of the Efimov effect in a simple model
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2023
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| _version_ | 1866909797951995904 |
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| author | Fermi, Davide Ferretti, Daniele Teta, Alessandro |
| author_facet | Fermi, Davide Ferretti, Daniele Teta, Alessandro |
| contents | We consider a system of three identical bosons in $\mathbb{R}^3$ with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a>0$. Using a quadratic form approach we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of $a$. In particular this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, \emph{i.e.}, we show that the Hamiltonian has an infinite sequence of negative eigenvalues $E_n$ accumulating at zero and fulfilling the asymptotic geometrical law $\;E_{n+1} / E_n \; \to \; e^{-\frac{2π}{s_0}}\,\; \,\text{for} \,\; n\to +\infty$ holds, where $s_0\approx 1.00624$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_12157 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rigorous derivation of the Efimov effect in a simple model Fermi, Davide Ferretti, Daniele Teta, Alessandro Mathematical Physics Quantum Gases Quantum Physics 81Q10, 81Q15, 70F07, 46N50 We consider a system of three identical bosons in $\mathbb{R}^3$ with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a>0$. Using a quadratic form approach we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of $a$. In particular this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, \emph{i.e.}, we show that the Hamiltonian has an infinite sequence of negative eigenvalues $E_n$ accumulating at zero and fulfilling the asymptotic geometrical law $\;E_{n+1} / E_n \; \to \; e^{-\frac{2π}{s_0}}\,\; \,\text{for} \,\; n\to +\infty$ holds, where $s_0\approx 1.00624$. |
| title | Rigorous derivation of the Efimov effect in a simple model |
| topic | Mathematical Physics Quantum Gases Quantum Physics 81Q10, 81Q15, 70F07, 46N50 |
| url | https://arxiv.org/abs/2306.12157 |