Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914047525388288 |
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| author | Ferretti, Daniele Teta, Alessandro |
| author_facet | Ferretti, Daniele Teta, Alessandro |
| contents | We discuss some properties of a model Hamiltonian for a system of three bosons interacting via zero-range forces in three dimensions. In order to avoid the well known instability phenomenon, we consider the so-called Minlos-Faddeev regularization of such Hamiltonian, heuristically corresponding to the introduction of a three-body repulsion. We review the main concerning results recently obtained. In particular, starting from a suitable quadratic form $Q$, the self-adjoint and bounded from below Hamiltonian $\mathcal H$ can be constructed provided that the strength $γ$ of the three-body force is larger than a threshold parameter $γ_c$. Moreover, we give an alternative and much simpler proof of the above result whenever $γ> γ'_c$, with $γ'_c$ strictly larger than $γ_c$. Finally, we show that the threshold value $γ_c$ is optimal, in the sense that the quadratic form $Q$ is unbounded from below if $γ<γ_c$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_00313 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions Ferretti, Daniele Teta, Alessandro Mathematical Physics Quantum Physics 81Q10, 81Q15, 70F07, 46N50, We discuss some properties of a model Hamiltonian for a system of three bosons interacting via zero-range forces in three dimensions. In order to avoid the well known instability phenomenon, we consider the so-called Minlos-Faddeev regularization of such Hamiltonian, heuristically corresponding to the introduction of a three-body repulsion. We review the main concerning results recently obtained. In particular, starting from a suitable quadratic form $Q$, the self-adjoint and bounded from below Hamiltonian $\mathcal H$ can be constructed provided that the strength $γ$ of the three-body force is larger than a threshold parameter $γ_c$. Moreover, we give an alternative and much simpler proof of the above result whenever $γ> γ'_c$, with $γ'_c$ strictly larger than $γ_c$. Finally, we show that the threshold value $γ_c$ is optimal, in the sense that the quadratic form $Q$ is unbounded from below if $γ<γ_c$. |
| title | Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions |
| topic | Mathematical Physics Quantum Physics 81Q10, 81Q15, 70F07, 46N50, |
| url | https://arxiv.org/abs/2207.00313 |