Open analytic problems in the mutual-distance (angle-independent) reduction of the quantum three-body problem: polynomial spectral theory, self-adjointness, and unequal-mass (quasi)-exact solvability

For a quantum three-body system in 3D space whose pair interaction depends only on the mutual distances $r_{12},r_{13},r_{23}$, Turbiner, Miller and Escobar-Ruiz (J. Phys. A 50, 215201 (2017)) showed that the ground state and other angle-independent states are governed by a Schrödinger-type operator on the three-dimensional space of mutual distances: a Laplace–Beltrami kinetic energy on a curved manifold with metric determinant $D=6\tau_1(4\tau_2-\tau_1^2)$ (where $\tau_1,\tau_2,\tau_3$ are the elementary symmetric polynomials in $\rho_{ij}=r_{ij}^2$) plus an effective potential, with hidden algebra $sl(4,\mathbb{R})$ in the $\rho_{ij}$ variables and $h^{(3)}$ in the $\tau$ variables. They constructed an exactly-solvable generalized harmonic-oscillator potential and a quasi-exactly-solvable sextic potential whose eigenfunctions are a Gaussian-type factor $\Psi_0$ times polynomials in the $\tau$ variables. The announced generalization to arbitrary spatial dimension $d$ was subsequently published and is resolved; three precisely stated threads remain open: (i) the two- and three-variable orthogonal-polynomial families generated by the reduced operators (orthogonality measure, recurrence relations, completeness) are unstudied; (ii) no rigorous self-adjoint realization of the constructed (quasi)-exactly-solvable reduced Hamiltonians or classification of admissible parameter values exists; (iii) no classification of (quasi)-exactly-solvable potentials and invariant subspaces exists for unequal masses.

Background

The non-relativistic quantum three-body problem with pairwise interactions is, after separating the center of mass, a six-dimensional Schrödinger problem. When the interaction depends only on the mutual distances $r_{ij}=|\mathbf{r}_i-\mathbf{r}_j|$, a physically important subclass of states — the ground state and, more generally, all angle-independent ($S$-type) states — depends only on the three distances $r_{12},r_{13},r_{23}$. Turbiner, Miller and Escobar-Ruiz (J. Phys. A 50, 215201 (2017)) made this observation operational: splitting the relative-motion space into distances and angles, $\tfrac{1}{2}\Delta_r^{(6)}=\Delta_R(r_{ij})+\tilde\Delta(r_{ij},\Omega,\partial_\Omega)$, the angular part annihilates any angle-independent function, so these states obey a closed spectral problem on the space of relative distances — the set of triples $(r_{12},r_{13},r_{23})$ satisfying the triangle inequalities. After a gauge rotation, the distance-space kinetic energy $\Delta_R$ becomes a Laplace–Beltrami operator (the natural second-order differential operator of a curved Riemannian space, here non-conformally-flat) with metric $g_{\mu\nu}$ of determinant $D=6\tau_1(4\tau_2-\tau_1^2)$, where $\tau_1=\rho_{12}+\rho_{13}+\rho_{23}$, $\tau_2=\rho_{12}\rho_{13}+\rho_{12}\rho_{23}+\rho_{13}\rho_{23}$, $\tau_3=\rho_{12}\rho_{13}\rho_{23}$ are the elementary symmetric polynomials in $\rho_{ij}=r_{ij}^2$. The determinant vanishes at degenerate triangles (the boundary of the configuration domain), and an effective potential $\tilde V$ arising from the gauge rotation is singular there; the reduced Hamiltonian is Hermitian with measure $D^{-1/2}$.

In the $\rho_{ij}$ variables the gauge-rotated kinetic energy is an algebraic (preserving polynomial spaces) differential operator with a hidden Lie algebra $sl(4,\mathbb{R})$ — that is, it can be rewritten entirely in terms of first-order differential operators generating $sl(4,\mathbb{R})$ — while in the symmetric $\tau$ variables it has the hidden algebra $h^{(3)}$ (the algebra of first-order operators preserving the polynomial ring in three variables). A quantum system is called exactly solvable (ES) if infinitely many eigenstates can be obtained algebraically, and quasi-exactly solvable (QES) if only finitely many can (the algebraic operator then preserves a flag of finite-dimensional polynomial spaces $\mathcal{P}_N^{(1,2,3)}\subset\mathcal{P}_N^{(1,2)}\subset\mathcal{P}_N^{(1)}$ of dimension $\sim N^3,\sim N^2,\sim N$). The paper constructs an ES generalized harmonic-oscillator potential and a QES sextic polynomial potential, both admitting an extra integral, whose eigenfunctions factorize as $\mathrm{Pol}_N\cdot\Psi_0$ with

$\Psi_0=\tau_1^{1/4}(4\tau_2-\tau_1^2)^{\gamma/2}\,e^{-\omega\tau_1-\frac{A}{2}\tau_1^2},$

with parameters $\gamma$, $\omega>0$, $A\ge 0$. At $A=0$ the polynomial factors are mutually orthogonal with respect to $\Psi_0^2$ on the distance domain; the one-variable subfamily in $\tau_1$ is the classical Laguerre family, while the two- and three-variable families were stated by the authors to be unstudied in the literature. The reduced problem also possesses a first-order symmetry operator $L_1=(\rho_{13}-\rho_{23})\partial_{\rho_{12}}+(\rho_{23}-\rho_{12})\partial_{\rho_{13}}+(\rho_{12}-\rho_{13})\partial_{\rho_{23}}$ which, in suitable coordinates $(w_1,w_2,w_3)$, is simply $\partial_{w_3}$ and separates one variable; however, $L_1$ is a 'particular integral': it commutes with the reduced (distance-space) Hamiltonian, but not with the full six-dimensional relative-motion Hamiltonian, whose angular part couples distances and angles (a terminology the authors use, citing Turbiner (J. Phys. A 46, 025203 (2013)), in the d-dimensional sequel, J. Math. Phys. 59, 022108 (2018), and in the four-body sequel, J. Math. Phys. 60, 062101 (2019)). An appendix shows that for unequal masses $m_1,m_2,m_3$ the distance-space operator keeps the same algebraic structure (with reduced masses $\mu_{ij}$) but lives on a different manifold, with a modified determinant $D_m$. The reduction matters computationally: it reduces ground-state and $S$-state calculations (e.g. for helium-like and small molecular systems) from six-dimensional to three-dimensional integrals in variational treatments.

The 2017 paper closed with the statement that the generalization to arbitrary spatial dimension $d$ 'will be presented elsewhere', together with remarks leaving the polynomial families uncharacterized, the self-adjointness of the reduced operators asserted only via a parenthetical note on choosing $\gamma$ so that boundary terms vanish, and the unequal-mass case algebraically formulated but not classified.

Problem Statement

Complete the analytic theory of the angle-independent (mutual-distance) reduction of the quantum three-body problem in 3D space with distance-dependent pair interactions, as constructed in Turbiner–Miller–Escobar-Ruiz (J. Phys. A 50, 215201 (2017)) and its d-dimensional sequel — that is, settle the three open threads that the source work left, whose common objective is to turn the formal (quasi)-exact-solvability construction on the space of relative distances into a full spectral and algebraic theory:

1. Polynomial spectral theory. For the exactly-solvable ($A=0$) and quasi-exactly-solvable ($A>0$) reduced operators in the symmetric variables $\tau_1,\tau_2,\tau_3$ (equivalently, the $\rho_{ij}$ or $(w_1,w_2,w_3)$ variables), characterize the polynomial eigenfunction families multiplying $\Psi_0$: determine their orthogonality measure on the distance domain (including the role of the curved measure $D^{-1/2}$ and of $\Psi_0^2$), derive their recurrence relations, and establish (or refute) their completeness/basis property in the corresponding $L^2$ space; in particular, treat the two- and three-variable families beyond the known one-variable Laguerre restriction.

2. Rigorous spectral theory. For the (quasi)-exactly-solvable reduced Hamiltonians constructed in the source work, $H_{rd}=-\Delta_{LB}+\tilde V+V$, where $V$ is the authors' generalized-harmonic-oscillator (ES) or sextic (QES) potential with parameters $(\gamma,\,N,\,\omega>0,\,A\ge 0)$, prove a self-adjoint realization on the distance domain (a degenerate elliptic operator whose metric determinant $D=6\tau_1(4\tau_2-\tau_1^2)$ vanishes at degenerate triangles), and classify the admissible parameter values — in particular $\gamma$ for given invariant-space degree $N$ — for which the boundary terms vanish, so that the (quasi)-exact eigenfunctions belong to the domain of a self-adjoint operator and the variational use of the reduction is justified. This determines, for the quasi-exactly-solvable Hamiltonian, whether a $\gamma$ can be chosen for given $N$ so that the boundary terms vanish and the Hamiltonian acts as a self-adjoint operator, and for which $N$ and $\gamma$ this holds.

3. Unequal masses. For arbitrary masses $m_1,m_2,m_3$, classify the (quasi)-exactly-solvable potentials and the finite-dimensional polynomial invariant subspaces of the non-equal-mass reduced operator — beyond the known exactly-solvable harmonic-oscillator case with its cubic correction terms and the established non-extendability of the pairwise sextic quasi-exactly-solvable oscillator — or prove no further non-trivial families exist.

All three items are required parts of the objective; within item 1, determining the orthogonality measure, deriving the recurrence relations, and establishing the completeness/basis property are each required, and no proper subset of them settles item 1.

(The generalization to arbitrary spatial dimension, also announced as open by the source, was published by the same authors in J. Math. Phys. 59, 022108 (2018) and is therefore not part of the problem.)

References

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2. A. V. Turbiner, W. Miller, Jr., M. A. Escobar-Ruiz, "Three-body problem in d-dimensional space: ground state, (quasi)-exact-solvability", J. Math. Phys. 59, 022108 (2018). DOI 10.1063/1.4994397; arXiv:1707.01324; https://arxiv.org/abs/1707.01324
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