Fermionic distance-only eigenstates and completeness of the algebraic spectrum for the quantum three-body problem in the space of relative distances

For three quantum particles in $d$-dimensional space with pair interactions depending only on the mutual distances $r_{ij}$, a family of eigenstates including the ground state depends on the three distances $r_{12},r_{13},r_{23}$ alone. Turbiner, Miller and Escobar-Ruiz reduced the spectral problem for these states to a three-dimensional Schrödinger operator on a curved configuration space with hidden Lie-algebraic structure ($sl(4,\mathbb{R})$ in squared-distance variables, the Heisenberg algebra $h(3)$ in geometric variables) and constructed exactly solvable (singular harmonic-type) and quasi-exactly solvable (singular sextic) potentials. Two central questions posed in that work remain open: (i) the conjecture that three fermions, and general $n$-body systems, likewise possess families of eigenstates, including the ground state, whose coordinate functions depend only on mutual distances; and (ii) whether the algebraic (polynomial) spectrum of the exactly solvable model is complete, i.e., whether it exhausts the bound-state spectrum of the reduced problem in the space of relative distances or misses non-algebraic bound states. As of 2026 the bosonic ground-state claim is proven for all $n$ in $d\ge n-1$, whereas the fermionic conjecture remains open and no study of spectral completeness is known.

Background

Consider three quantum particles of equal mass in $d$-dimensional space, interacting through a potential that depends only on the mutual (relative) distances $r_{ij}=|\mathbf{r}_i-\mathbf{r}_j|$ between particle pairs. After separating the center-of-mass motion, the spectral problem lives in the $2d$-dimensional space of relative motion, which splits into three radial coordinates — the sides $r_{12},r_{13},r_{23}$ of the so-called triangle of interaction — and $2d-3$ angular coordinates. A basic structural fact, emphasized by Turbiner, Miller and Escobar-Ruiz, is that such potentials possess a family of eigenstates, including the ground state, whose wavefunctions depend on the three mutual distances alone; these are the zero-total-angular-momentum, permutation-invariant (S-)states. (For the ground state of scalar particles this is a theorem: by rotational invariance and nondegeneracy the ground state must be a function of the $r_{ij}$, which form an integrity basis for translation- and rotation-invariant functions.)

The program of that paper (following its predecessor for $d=3$) makes this observation quantitative. In the squared distances $\rho_{ij}=r_{ij}^2$ the flat kinetic-energy operator becomes a second-order algebraic differential operator with a hidden $sl(4,\mathbb{R})$ Lie-algebra structure (the algebra of real $4\times4$ matrices, acting via first-order differential operators whose second-order envelope preserves a flag of finite-dimensional polynomial spaces). In the elementary symmetric polynomials of the $\rho_{ij}$ — the geometric variables $P=\rho_{12}+\rho_{13}+\rho_{23}$ (a moment-of-inertia type variable), $S$ (proportional to the squared area of the interaction triangle), and $T=\rho_{12}\rho_{13}\rho_{23}$ (its volume-type companion) — the hidden algebra is the Heisenberg algebra $h(3)$, the same algebra that governs the three-particle Calogero model (the integrable one-dimensional system of particles interacting through inverse-square potentials). After a gauge rotation (a similarity transformation of the operator by a prefactor built from the ground-state asymptotics), the radial Hamiltonian becomes a Laplace–Beltrami operator (the covariant second-derivative operator of a curved space) plus an effective potential: it describes a three-dimensional quantum particle in a curved space with a $d$-independent metric and a $d$-dependent singular effective potential, degenerating at $d=1$ to a flat two-dimensional problem.

Within this framework the authors construct an exactly solvable (ES) model — an $S_3$-invariant ($S_3$ being the group of permutations of the three particles) generalized harmonic oscillator with singular terms, whose polynomial eigenstates times the gauge factor form an infinite algebraic family — and quasi-exactly solvable (QES) sextic potentials, for which only finite-dimensional invariant polynomial spaces (of dimensions growing as $\sim N^3$, $\sim N^2$, and $\sim N$ in the three-, two-, and one-variable sectors) survive. Here exactly solvable means that infinitely many eigenstates are obtained by algebraic means, and quasi-exactly solvable that only a finite algebraic sector is available, in the standard sense of Turbiner's classification. The authors state two explicit gaps in this solvability theory. First, a conjecture: that for three fermions, and more generally for $n$-body systems, there exists a family of eigenstates, including the ground state, whose coordinate functions depend only on mutual distances. Second, a disclaimer about the ES construction in the geometric variables $P,S,T$: 'We do not know whether their spectra is complete.' This completeness question concerns the reduced problem in the space of relative distances: for the original problem in the full relative-motion space the algebraic sector is manifestly incomplete, since infinitely many angle-dependent eigenfunctions exist and are, as the authors note, likely of non-algebraic nature. A precise formulation of the fermionic conjecture requires care. The source works with spinless particles, for which the permutation antisymmetry demanded by fermionic statistics must reside in the spatial wavefunction itself. Since the coordinates are the labeled distances $r_{12},r_{13},r_{23}$, a function of the distances alone need not be permutation-symmetric: each transposition of two particles exchanges two labeled distances and fixes the third, so, for example, the Vandermonde product $\Delta=(r_{12}-r_{13})(r_{12}-r_{23})(r_{13}-r_{23})$, and $\Delta$ multiplied by any totally symmetric function of the distances, are nonzero totally antisymmetric functions depending on the distances alone. The conjecture is therefore well posed — and nontrivial — in its spinless reading, namely the existence of alternating, distance-only eigenstate families; readings in which antisymmetry is carried by spin degrees of freedom are possible variants but do not capture the source's spinless setting.

Problem Statement

Establish the exact-solvability status of the quantum three-body problem in the space of relative distances, by resolving the two questions left open by Turbiner, Miller and Escobar-Ruiz (J. Math. Phys. 59, 022108 (2018)):

1. Fermionic and $n$-body distance-only conjecture. Prove or refute the conjecture, in both the three-fermion case and the source's general $n$-body generalization with pair interactions depending only on mutual distances, that there exists a family of eigenstates of the Hamiltonian, including the ground state, whose coordinate functions depend only on the mutual distances $\{r_{ij}\}$. The source states the conjecture for spinless particles, for which the fermionic antisymmetry must reside in the spatial wavefunction itself; because the coordinates are the labeled distances, alternating distance-only functions do exist (for instance, the Vandermonde product $\Delta=(r_{12}-r_{13})(r_{12}-r_{23})(r_{13}-r_{23})$ times any totally symmetric function of the distances), so this spinless, alternating formulation is well posed and is the reading a resolution must address: either construct the eigenstate families together with the reduced differential operator in the space of relative distances for which they are eigenfunctions, or prove that no such family exists beyond the bosonic case. A formulation in which the antisymmetry is carried by spin degrees of freedom does not settle the conjecture on its own, since for a spin-independent Hamiltonian the spatial factor then obeys the bosonic problem, whose distance-only ground state is already established (see Background).

2. Completeness of the algebraic spectrum. For the exactly solvable, $S_3$-invariant, singular harmonic-oscillator-type potential constructed in the geometric variables $(P,S,T)$ — the model to which the source's disclaimer 'We do not know whether their spectra is complete' is attached — determine whether the algebraic (polynomial) eigenstates exhaust the bound-state spectrum of the reduced Hamiltonian on the space of relative distances, or whether additional non-algebraic bound states exist there.

These two questions jointly determine the solvability claim of the program: question 1 fixes which particle statistics the distance-only reduction captures, and question 2 fixes how much of the reduced spectrum the algebraic construction actually recovers.

References

1. Alexander V. Turbiner, Willard Miller, 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
2. Alexander V Turbiner, Willard Miller, Adrian M Escobar-Ruiz, 'Three-body problem in 3D space: ground state, (quasi)-exact-solvability', J. Phys. A: Math. Theor. 50, 215201 (2017). DOI: 10.1088/1751-8121/aa6cc2; arXiv:1611.08157; https://arxiv.org/abs/1611.08157
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