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arXiv:2204.11997 · Journal of Experimental and Theoretical Physics Letters

Minlos–Faddeev Regularization of Zero-Range Interactions in the Three-Body Problem

O. I. Kartavtsev, A. V. Malykh

Published Jul 25, 2022• 3 citations

Abstract

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 $$\sigma < {{\sigma }_{c}}$$ , the additional boundary conditions of two types should be imposed at the triple-collision point for $${{\sigma }_{c}} \leqslant \sigma < {{\sigma }_{e}}$$ and $${{\sigma }_{e}} < \sigma < {{\sigma }_{r}}$$ , and the problem is regularized for $$\sigma \geqslant {{\sigma }_{r}}$$ . Critical values $${{\sigma }_{c}} < {{\sigma }_{e}} < {{\sigma }_{r}}$$ separating different alternatives are determined both for a two-component three-body system and for three identical bosons.

DOI: 10.1134/s002136402260118x

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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

\sigma < {{\sigma }_{c}}

, the additional boundary conditions of two types should be imposed at the triple-collision point for

{{\sigma }_{c}} \leqslant \sigma < {{\sigma }_{e}}

and

{{\sigma }_{e}} < \sigma < {{\sigma }_{r}}

, and the problem is regularized for

\sigma \geqslant {{\sigma }_{r}}

. Critical values

{{\sigma }_{c}} < {{\sigma }_{e}} < {{\sigma }_{r}}

separating different alternatives are determined both for a two-component three-body system and for three identical bosons.

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 consider a system of three identical bosons in

\mathbb {R}^3

R 3 with two-body zero-range interactions and a three-body hard-core repulsion of a given radius

a > 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

E_n

E n accumulating at zero and fulfilling the asymptotic geometrical law

\;E_{n+1} / E_n \; \rightarrow \; e^{-\frac{2\pi }{s_0}}\,\; \,\text {for} \,\; n\rightarrow +\infty

E n + 1 / E n → e - 2 π s 0 for n → + ∞ holds, where

s_0\approx 1.00624

s 0 ≈ 1.00624 .

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.