Three-Body Hamiltonian with Regularized Zero-Range Interactions in Dimension Three

Abstract

We study the Hamiltonian for a system of three identical bosons in dimension three interacting via zero-range forces. In order to avoid the fall to the center phenomenon emerging in the standard Ter-Martirosyan--Skornyakov (TMS) Hamiltonian, known as Thomas effect, we develop in detail a suggestion given in a seminal paper of Minlos and Faddeev in 1962 and we construct a regularized version of the TMS Hamiltonian which is self-adjoint and bounded from below. The regularization is given by an effective three-body force, acting only at short distance, that reduces to zero the strength of the interactions when the positions of the three particles coincide. The analysis is based on the construction of a suitable quadratic form which is shown to be closed and bounded from below. Then, domain and action of the corresponding Hamiltonian are completely characterized and a regularity result for the elements of the domain is given. Furthermore, we show that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled non local interactions, also called separable potentials, with a suitably renormalized coupling constant.

arXiv comment45 pages, minor changes, references added

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Cited by in corpus (8)

• From Short-Range to Contact Interactions in Two-dimensional Many-Body Quantum Systems
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• Rigorous derivation of the Efimov effect in a simple model
• Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions
• On the Weakness of Short-Range Interactions in Fermi Gases
• Mass-ratio condition for non-binding of three two-component particles with contact interactions
• Hamiltonians for Quantum Systems with Contact Interactions
• Efimov spectrum in the Born--Oppenheimer picture of 2+1 system with zero-range heavy-light interactions