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

This article is cited in 11 scientific papers (total in 11 papers)

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).


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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
Received: 17.04.2014

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
\by R.~A.~Minlos
\paper A system of three quantum~particles with point-like interactions
\jour Uspekhi Mat. Nauk
\yr 2014
\vol 69
\issue 3(417)
\pages 145--172
\jour Russian Math. Surveys
\yr 2014
\vol 69
\issue 3
\pages 539--564

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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  crossref  adsnasa  isi  elib  scopus
    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.  crossref  mathscinet  zmath  isi  scopus
    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.  crossref  mathscinet  zmath  isi  scopus
    4. T. Moser, R. Seiringer, “Stability of a fermionic $N+1$ particle system with point interactions”, Comm. Math. Phys., 356:1 (2017), 329–355  crossref  mathscinet  zmath  isi  scopus
    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  mathscinet  zmath  isi
    6. G. Basti, A. Teta, “Efimov effect for a three-particle system with two identical fermions”, Ann. Henri Poincaré, 18:12 (2017), 3975–4003  crossref  mathscinet  zmath  isi  scopus
    7. A. Michelangeli, A. Ottolini, “On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians”, Rep. Math. Phys., 79:2 (2017), 215–260  crossref  mathscinet  zmath  isi
    8. K. Yoshitomi, “Finiteness of the discrete spectrum in a three-body system with point interaction”, Math. Slovaca, 67:4 (2017), 1031–1042  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  isi
    10. Moser T., Seiringer R., “Stability of the 2+2 Fermionic System With Point Interactions”, Math. Phys. Anal. Geom., 21:3 (2018), 19  crossref  mathscinet  zmath  isi  scopus
    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  crossref  isi
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