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.