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 Uspekhi Mat. Nauk, 2014, Volume 69, Issue 3(417), Pages 145–172 (Mi umn9589)

A system of three quantum particles with point-like interactions

R. A. Minlos

A. A. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: Consider a quantum three-particle system consisting of two fermions of unit mass and another particle of mass $m>0$ interacting in a point-like manner with the fermions. Such systems are studied here using the theory of self-adjoint extensions of symmetric operators: the Hamiltonian of the system is constructed as an extension of the symmetric energy operator
$$H_0=-\frac{1}{2}(\frac{1}{m}\Delta_y+\Delta_{x_1}+\Delta_{x_2}),$$
which is defined on the functions in $L_2(\mathbb{R}^3)\otimes L_2^{\operatorname{asym}}(\mathbb{R}^3\times\mathbb{R}^3)$ that vanish whenever the position of the third particle coincides with the position of a fermion. To construct a natural family of extensions of $H_0$, one must solve the problem of self-adjoint extensions for an auxiliary sequence $\{T_l, l=0,1,2,…\}$ of symmetric operators acting in $L_2(\mathbb{R}^3)$. All the operators $T_l$ with even $l$ are self-adjoint, and for every odd $l$ there are two numbers $0<m_l^{(1)}<m_l^{(2)}<\infty$ such that $T_l$ is self-adjoint and lower semibounded for $m>m_l^{(2)}$, and has deficiency indices for $m\leqslant m_l^{(2)}$. When $m\in[m_l^{(1)}, m_l^{(2)}]$, every self-adjoint extension of $T_l$ which is invariant under rotations of $\mathbb{R}^3$ is lower semibounded, but if $0<m<m_l^{(1)}$, then it has an infinite sequence of eigenvalues $\{\lambda_n\}$ of multiplicity $2l+1$ such that $\lambda_n\to-\infty$ as $n\to\infty$ (the Thomas effect). It follows from the last fact that there is a sequence of bound states of $H_0$ with spectrum $P^2/(2(m+2))+z_n$, where the numbers $z_n<0$ cluster at 0 (Efimov's effect).
Bibliography: 19 titles.

Keywords: symmetric operator, deficiency indices, semibounded operator, self-adjoint extensions, spectrum, Mellin transform, the Riemann–Hilbert–Privalov problem.

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-12410 This paper was written with the support of the Russian Foundation for Basic Research (grant no. 13-01-12410).

DOI: https://doi.org/10.4213/rm9589

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English version:
Russian Mathematical Surveys, 2014, 69:3, 539–564

Bibliographic databases:

Document Type: Article
UDC: 517.958:530.145+517.984
MSC: 81Q10, 81V15

Citation: R. A. Minlos, “A system of three quantum particles with point-like interactions”, Uspekhi Mat. Nauk, 69:3(417) (2014), 145–172; Russian Math. Surveys, 69:3 (2014), 539–564

Citation in format AMSBIB
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This publication is cited in the following articles:
1. M. Correggi, D. Finco, A. Teta, “Energy lower bound for the unitary $N+1$ fermionic model”, EPL, 111:1 (2015), 10003
2. M. Correggi, G. Dell'Antonio, D. Finco, A. Michelangeli, A. Teta, “A class of Hamiltonians for a three-particle fermionic system at unitarity”, Math. Phys. Anal. Geom., 18:1 (2015), 32, 36 pp.
3. A. Michelangeli, P. Pfeiffer, “Stability of the $(2+2)$-fermionic system with zero-range interaction”, J. Phys. A, 49:10 (2016), 105301, 27 pp.
4. T. Moser, R. Seiringer, “Stability of a fermionic $N+1$ particle system with point interactions”, Comm. Math. Phys., 356:1 (2017), 329–355
5. G. Basti, A. Teta, “On the quantum mechanical three-body problem with zero-range interactions”, Functional analysis and operator theory for quantum physics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2017, 71–93
6. G. Basti, A. Teta, “Efimov effect for a three-particle system with two identical fermions”, Ann. Henri Poincaré, 18:12 (2017), 3975–4003
7. A. Michelangeli, A. Ottolini, “On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians”, Rep. Math. Phys., 79:2 (2017), 215–260
8. K. Yoshitomi, “Finiteness of the discrete spectrum in a three-body system with point interaction”, Math. Slovaca, 67:4 (2017), 1031–1042
9. A. Michelangeli, A. Ottolini, “Multiplicity of self-adjoint realisations of the $(2+1)$-fermionic model of Ter-Martirosyan-Skornyakov type”, Rep. Math. Phys., 81:1 (2018), 1–38
10. Moser T., Seiringer R., “Stability of the 2+2 Fermionic System With Point Interactions”, Math. Phys. Anal. Geom., 21:3 (2018), 19
11. Becker S., Michelangeli A., Ottolini A., “Spectral Analysis of the 2+1 Fermionic Trimer With Contact Interactions”, Math. Phys. Anal. Geom., 21:4 (2018), 35
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