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arXiv:2107.07188 · Annales Henri Poincaré

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

Giulia Basti, Claudio Cacciapuoti, Domenico Finco, Alessandro Teta

Published Jul 10, 2022• 13 citations
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Abstract

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.

DOI
10.1007/s00023-022-01214-9

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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.
Abstract We study the Hamiltonian for a system of N identical bosons interacting with an impurity, i.e., a different particle, via zero-range forces in dimension three. It is well known that, following the standard approach, one obtains the Ter-Martirosyan Skornyakov Hamiltonian which is unbounded from below. In order to avoid such instability problem, we introduce a three-body force acting at short distances. The effect of this force is to reduce to zero the strength of the zero-range interaction between two particles, i.e., the impurity and a boson, when another boson approaches the common position of the first two particles. We show that the Hamiltonian defined with such regularized interaction is self-adjoint and bounded from below if the strength of the three-body force is sufficiently large. The method of the proof is based on a careful analysis of the corresponding quadratic form.
We study the stability problem for a non-relativistic quantum system in dimension three composed by N ≥ 2 identical fermions, with unit mass, interacting with a different particle, with mass m, via a zero-range interaction of strength α ∈ ℝ. We construct the corresponding renormalized quadratic (or energy) form [Formula: see text] and the so-called Skornyakov–Ter–Martirosyan symmetric extension H α , which is the natural candidate as Hamiltonian of the system. We find a value of the mass m*(N) such that for m > m*(N) the form [Formula: see text] is closed and bounded from below. As a consequence, [Formula: see text] defines a unique self-adjoint and bounded from below extension of H α and therefore the system is stable. On the other hand, we also show that the form [Formula: see text] is unbounded from below for m < m*(2). In analogy with the well-known bosonic case, this suggests that the system is unstable for m < m*(2) and the so-called Thomas effect occurs.
We consider a quantum system in dimension three composed by a group of NNN identical fermions, with mass 1/2, interacting via zero-range interaction with a group of MMM identical fermions of a different type, with mass m/2m/2m/2. Exploiting a renormalization procedure, we construct the corresponding quadratic (or energy) form and define the so-called Ter-Martirosyan-Skornyakov extension HαH_αHα​, which is the natural candidate as a possible Hamiltonian of the system. In the particular case M=1, under a suitable condition on the parameters mmm, NNN, we show that the quadratic form is unbounded from below. In the same setting we prove that HαH_αHα​ is not a self-adjoint and bounded from below operator and this in particular suggests that the so-called Thomas effect could occur.
To regularize the three-body problem, Minlos and Faddeev suggested a modification of zero-range model, which diminishes interaction at the triple-collision point. The analysis reveals that this regularization results in four alternatives depending on the regularization parameter σ. Explicitly, Efimov or Thomas effects remain for σ<σc\sigma < {{\sigma }_{c}}σ<σc​ , the additional boundary conditions of two types should be imposed at the triple-collision point for σc⩽σ<σe{{\sigma }_{c}} \leqslant \sigma < {{\sigma }_{e}}σc​⩽σ<σe​ and σe<σ<σr{{\sigma }_{e}} < \sigma < {{\sigma }_{r}}σe​<σ<σr​ , and the problem is regularized for σ⩾σr\sigma \geqslant {{\sigma }_{r}}σ⩾σr​ . Critical values σc<σe<σr{{\sigma }_{c}} < {{\sigma }_{e}} < {{\sigma }_{r}}σc​<σe​<σr​ separating different alternatives are determined both for a two-component three-body system and for three identical bosons.
We consider a Hamiltonian describing three quantum particles in dimension one interacting through two-body short-range potentials. We prove that, as a suitable scale parameter in the potential terms goes to zero, such a Hamiltonian converges to one with zero-range (also called delta or point) interactions. The convergence is understood in the norm resolvent sense. The two-body rescaled potentials are of the form vσε(xσ)=ε−1vσ(ε−1xσ), where σ = 23, 12, 31 is an index that runs over all the possible pairings of the three particles, xσ is the relative coordinate between two particles, and ε is the scale parameter. The limiting Hamiltonian is the one formally obtained by replacing the potentials vσ with ασδσ, where δσ is the Dirac delta-distribution centered on the coincidence hyperplane xσ = 0 and ασ=∫Rvσdxσ. To prove the convergence of the resolvents, we make use of Faddeev’s equations.
Abstract We consider a system of three identical bosons in R3\mathbb {R}^3R3 R 3 with two-body zero-range interactions and a three-body hard-core repulsion of a given radius a>0 a > 0a>0 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, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues EnE_nEn​ E n accumulating at zero and fulfilling the asymptotic geometrical law   En+1/En  →  e−2πs0    for   n→+∞\;E_{n+1} / E_n \; \rightarrow \; e^{-\frac{2\pi }{s_0}}\,\; \,\text {for} \,\; n\rightarrow +\infty En+1​/En​→e−s0​2π​forn→+∞ E n + 1 / E n → e - 2 π s 0 for n → + ∞ holds, where s0≈1.00624s_0\approx 1.00624s0​≈1.00624 s 0 ≈ 1.00624 .