A system of three quantum particles with point-like interactions

R. A. Minlos

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 69 (2014) no. 3, pp. 539-564

Résumé

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 $$ H_0=-\frac{1}{2}\biggl(\frac{1}{m}\Delta_y+\Delta_{x_1}+\Delta_{x_2}\biggr), $$ 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 $\{T_l,\ l=0,1,2,\dots\}$ 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 $0$ 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 $0$, 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 $z_n0$ cluster at 0 (Efimov's effect). Bibliography: 19 titles.

Détail

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

Bibliographie

[1] F. A. Berezin, L. D. Faddeev, “Remark on the Schrödinger equation with singular potential”, Soviet Math. Dokl., 2 (1961), 372–375 | MR | Zbl

[2] R. A. Minlos, L. D. Faddeev, “On the point interaction for a three-particle system in quantum mechanics”, Soviet Physics Dokl., 6 (1962), 1072–1074 | MR

[3] R. A. Minlos, L. D. Faddeev, “Comment on the problem of three particles with point interactions”, Soviet Physics JETP, 14 (1962), 1315–1316 | MR

[4] R. A. Minlos, “On the point interaction of three particles”, Applications of selfadjoint extensions in quantum physics (Dubna, 1987), Lecture Notes in Phys., 234, Springer, Berlin, 1989, 138–145 | DOI | MR | Zbl

[5] R. A. Minlos, M. Kh. Shermatov, “Point interaction of three particles”, Moscow Univ. Math. Bull., 44:6 (1989), 7–15 | MR | Zbl

[6] M. Kh. Shermatov, “Point interaction between two fermions and one particle of a different nature”, Theoret. and Math. Phys., 136:2 (2003), 1119–1130 | DOI | DOI | MR | Zbl

[7] A. M. Mel'nikov, R. A. Minlos, “On the pointlike interaction of three different particles”, Many-particle Hamiltonians: spectra and scattering, Adv. Soviet Math., 5, Amer. Math. Soc., Providence, RI, 1991, 99–112 | MR | Zbl

[8] R. A. Minlos, “On point-like interaction between $n$ fermions and another particle”, Mosc. Math. J., 11:1 (2011), 113–127 | MR | Zbl

[9] R. A. Minlos, “Remark on my paper `On point-like interaction between $n$ fermions and another particle' ”, Mosc. Math. J., 11:4 (2011), 815–817 | MR | Zbl

[10] R. A. Minlos, “On pointlike interaction between three particles: two fermions and another particle”, ISRN Math. Phys., 2012 (2012), 230245, 18 pp. | DOI | Zbl

[11] R. Minlos, “On point-like interaction of three particles: two fermions and another particle. II”, Mosc. Math. J., 14:3 (2014), 617–637

[12] G. V. Skornyakov, K. A. Ter-Martirosyan, “Three body problem for short range forces. I. Scattering of low energy neutrons by deuterons”, Soviet Physics JETP, 4 (1957), 648–661 | MR | Zbl

[13] N. I. Akhiezer, I. M. Glazman, Theory of linear operators in Hilbert space, v. I, II, Frederick Ungar Publishing Co., New York, 1961, 1963, xi+147 pp., v+218 pp. | MR | MR | MR | Zbl | Zbl

[14] M. A. Naimark, Linear differential operators, v. I, II, Frederick Ungar Publishing Co., New York, 1967, 1968, xiii+144 pp., xv+352 pp. | MR | MR | Zbl | Zbl

[15] M. Sh. Birman, “K teorii samosopryazhennykh rasshirenii polozhitelno opredelennykh operatorov”, Matem. sb., 38(80):4 (1956), 431–450 | MR | Zbl

[16] I. M. Gel'fand, R. A. Minlos, Z. Ya. Shapiro, Representations of the rotation and Lorentz groups and their applications, Pergamon Press, Oxford–London–New York–Paris, 1963, xviii+366 pp. | MR | Zbl

[17] M. A. Lawrentjew, B. W. Schabat, Methoden der komplexen Funktionentheorie, Mathematik für Naturwissenschaft und Technik, 13, VEB Deutscher Verlag der Wissenschaften, Berlin, 1967, x+846 pp. | MR | MR | Zbl | Zbl

[18] E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Clarendon Press, Oxford, 1937, x+390 pp. | MR | Zbl

[19] R. E. A. C. Paley, N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Colloq. Publ., 19, Amer. Math. Soc., New York, 1934, viii+184 pp. | MR | MR | Zbl | Zbl