The quantum n-body problem in dimension $d\ge n-1$: ground state

Abstract

We employ generalized Euler coordinates for the $n$ body system in $d\ge n-1$ dimensional space, which consists of the centre-of-mass vector, relative (mutual), mass-independent distances $r_{ij}$ and angles as remaining coordinates. We prove that the kinetic energy of the quantum $n$-body problem for $d\ge n-1$ can be written as the sum of three terms: (i) kinetic energy of centre-of-mass, (ii) the second order differential operator ${\Delta }_{rad}$ which depends on relative distances alone and (iii) the differential operator $\Omega$ which annihilates any angle-independent function. The operator ${\Delta }_{rad}$ has a large reflection symmetry group $Z_2^{\oplus \frac{n(n-1)}{2}}$ and in ${\rho }_{ij}=r_{ij}^2$ variables is an algebraic operator, which can be written in terms of generators of their {\it hidden} algebra $sl(\frac{n(n-1)}{2}+1,R)$. Thus, ${\Delta }_{rad}$ makes sense of the Hamiltonian of a quantum Euler-Arnold $sl(\frac{n(n-1)}{2}+1,R)$ top in a constant magnetic field. It is conjectured that for any $n$, the similarity-transformed ${\Delta }_{rad}$ is the Laplace-Beltrami operator plus (effective) potential; thus, it describes a $\frac{n(n-1)}{2}$-dimensional quantum particle in curved space. This was verified for $n=2,3,4$. After de-quantization the similarity-transformed ${\Delta }_{rad}$ becomes the Hamiltonian of the classical top with variable tensor of inertia in an external potential. This approach allows a reduction of the $dn$-dimensional spectral problem to a $\frac{n(n-1)}{2}$ -dimensional spectral problem if the eigenfunctions depend only on relative distances. We prove that the ground state function of the $n$ body problem depends on relative distances alone.

32 pages, Theorem on ground state function added, as well as explicit formula for volume element, a few typos corrected

References in corpus (1)

• Three-body problem in $d$-dimensional space: ground state, (quasi)-exact-solvability

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