Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Four-body (an)harmonic oscillator in $d$-dimensional space: $S$-states,
  (quasi)-exact-solvability, hidden algebra $sl(7)$   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 $\mathbb{R}^d$ ($d\geq 3$) 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.

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The categories of this paper are: stat
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex