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Register a new userThree-body problem in $d$-dimensional space: ground state, (quasi)-exact-solvability
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As a straightforward generalization and extension of our previous paper, J. Phys. A50 (2017) 215201 we study aspects of the quantum and classical dynamics of a $3$-body system with equal masses, each body with $d$ degrees of freedom, with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories (which are in the interaction plane) in the classical case are of this type. It corresponds to a three-dimensional quantum particle moving in a curved space with special $d$-dimension-independent metric in a certain $d$-dependent singular potential, while at $d=1$ it elegantly degenerates to a two-dimensional particle moving in flat space. It admits a description in terms of pure geometrical characteristics of the interaction triangle which is defined by the three relative distances. The kinetic energy of the system is $d$-independent, it has a hidden $sl(4,R)$ Lie (Poisson) algebra structure, alternatively, the hidden algebra $h^{(3)}$ typical for the $H_3$ Calogero model as in the $d=3$ case. We find an exactly-solvable three-body $S^3$-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable three-body sextic polynomial type potential with singular terms. For both models an extra first order integral exists. It is shown that a straightforward generalization of the 3-body (rational) Calogero model to $d>1$ leads to two primitive quasi-exactly-solvable problems. The extension to the case of non-equal masses is straightforward and is briefly discussed.
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Due to its great importance for applications, we generalize and extend the approach of our previous papers to study aspects of the quantum and classical dynamics of a $4$-body system with equal masses in {it $d$}-dimensional space with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. The ground state (and some other states) in the quantum case and some trajectories in the classical case are of this type. We construct the quantum Hamiltonian for which these states are eigenstates. For $d geq 3$, this describes a six-dimensional quantum particle moving in a curved space with special $d$-independent metric in a certain $d$-dependent singular potential, while for $d=1$ it corresponds to a three-dimensional particle and coincides with the $A_3$ (4-body) rational Calogero model; the case $d=2$ is exceptional and is discussed separately. The kinetic energy of the system has a hidden $sl(7,{bf R})$ Lie (Poisson) algebra structure, but for the special case $d=1$ it becomes degenerate with hidden algebra $sl(4,R)$. We find an exactly-solvable four-body $S_4$-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable four-body sextic polynomial type potential with singular terms. Naturally, the tetrahedron whose vertices correspond to the positions of the particles provides pure geometrical variables, volume variables, that lead to exactly solvable models. Their generalization to the $n$-body system as well as the case of non-equal masses is briefly discussed.
As a generalization and extension of our previous paper {it J. Phys. A: Math. Theor. 53 055302} cite{AME2020}, in this work we study a quantum 4-body system in $mathbb{R}^d$ ($dgeq 3$) with quadratic and sextic pairwise potentials in the {it relative distances}, $r_{ij} equiv {|{bf r}_i - {bf r}_j |}$, between particles. Our study is restricted to solutions in the space of relative motion with zero total angular momentum ($S$-states). In variables $rho_{ij} equiv r_{ij}^2$, the corresponding reduced Hamiltonian of the system possesses a hidden $sl(7;{bf R})$ Lie algebra structure. In the $rho$-representation it is shown that the 4-body harmonic oscillator with arbitrary masses and unequal spring constants is exactly-solvable (ES). We pay special attention to the case of four equal masses and to atomic-like (where one mass is infinite, three others are equal), molecular two-center (two masses are infinite, two others are equal) and molecular three-center (three infinite masses) cases. In particular, exact results in the molecular case are compared with those obtained within the Born-Oppenheimer approximation. The first and second order symmetries of non-interacting system are searched. Also, the reduction to the lower dimensional cases $d=1,2$ is discussed. It is shown that for four body harmonic oscillator case there exists an infinite family of eigenfunctions which depend on the single variable which is the moment-of-inertia of the system.
It is shown that the Confluent Heun Equation (CHEq) reduces for certain conditions of the parameters to a particular class of Quasi-Exactly Solvable models, associated with the Lie algebra $sl (2,{mathbb R})$. As a consequence it is possible to find a set of polynomial solutions of this quasi-exactly solvable version of the CHEq. These finite solutions encompass previously known polynomial solutions of the Generalized Spheroidal Equation, Razavy Eq., Whittaker-Hill Eq., etc. The analysis is applied to obtain and describe special eigen-functions of the quantum Hamiltonian of two fixed Coulombian centers in two and three dimensions.
It is shown that planar quantum dynamics can be related to 3-body quantum dynamics in the space of relative motion with a special class of potentials. As an important special case the $O(d)$ symmetry reduction from $d$ degrees of freedom to one degree is presented. A link between two-dimensional (super-integrable) systems and 3-body (super-integrable) systems is revealed. As illustration we present number of examples. We demonstrate that the celebrated Calogero-Wolfes 3-body potential has a unique property: two-dimensional quantum dynamics coincides with 3-body quantum dynamics on the line at $d=1$; it is governed by the Tremblay-Turbiner-Winternitz potential for parameter $k=3$.
We completely solve the problem of classifying all one-dimensional quantum potentials with nearest- and next-to-nearest-neighbors interactions whose ground state is Jastrow-like, i.e., of Jastrow type but depending only on differences of consecutive particles. In particular, we show that these models must necessarily contain a three-body interaction term, as was the case with all previously known examples. We discuss several particular instances of the general solution, including a new hyperbolic potential and a model with elliptic interactions which reduces to the known rational and trigonometric ones in appropriate limits.
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