Solution of the One-Dimensional N-Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials

Abstract

The quantum-mechanical problems of N 1-dimensional equal particles of mass m interacting pairwise via quadratic (``harmonical'') and/or inversely quadratic (``centrifugal'') potentials is solved. In the first case, characterized by the pair potential ¼mω2(xi − xj)2 + g(xi − xj)−2, g > −ℏ2/(4m), the complete energy spectrum (in the center-of-mass frame) is given by the formula E=ℏω(12N)12[12(N−1)+12N(N−1)(a+12)+ ∑ l=2Nlnl],with a = ½(1 + 4mgℏ−2)½. The N − 1 quantum numbers nl are nonnegative integers; each set {nl; l = 2, 3, ⋯, N} characterizes uniquely one eigenstate. This energy spectrum can also be written in the form Es = ℏω(½N)½ [½(N − 1) + ½N(N − 1)(a + ½) + s], s = 0, 2, 3, 4, ⋯, the multiplicity of the sth level being then given by the number of different sets of N − 1 nonnegative integers nl that are consistent with the condition s=∑l=2Nlnl. These equations are valid independently of the statistics that the particles satisfy, if g ≠ 0; for g = 0, the equations remain valid with a = ½ for Fermi statistics, a = −½ for Bose statistics. The eigenfunctions corresponding to these energy levels are not obtained explicitly, but they are rather fully characterized. A more general model is similarly solved, in which the N particles are divided in families, with the same quadratic interaction acting between all pairs, but with the inversely quadratic interaction acting only between particles belonging to the same family, with a strength that may be different for different families. The second model, characterized by the pair potential g(xi − xj)−2, g > −ℏ2/(4m), contains only scattering states. It is proved that an initial scattering configuration, characterized (in the phase space sector defined by the inequalities xi ≥ xi.1, i = 1, 2, ⋯, N = 1, to which attention may be restricted without loss of generality) by (initial) momenta pi, i = 1, 2, ⋯, N, goes over into a final configuration characterized uniquely by the (final) momenta pi′, with pi′=pN+1−i. This remarkably simple outcome is a peculiarity of the case with equal particles (i.e., equal masses and equal strengths of all pair potentials).

The tetrahexahedric angular Calogero model

“…As the potential is repulsive, the spectrum is positive semi-definite and continuous, 14) where the wave function depends on further (suppressed) quantum numbers, but q parametrizes the angular Hamiltonian eigenvalues [29] (see also the appendices of [2]),…”

“…The Calogero (or Calogero-Moser) model [1][2][3] 1 is the paradigmatical n-particle integrable system in one space dimension. Originally defined for the root system of A 1 ⊕ A n−1 , the Calogero model was quickly generalized for any finite Coxeter group of rank n [4].…”

LLL vs. LLM: Half BPS sector of SYM ≡ quantum Hall system

“…Let us first review the physical states of a finite QHS with arbitrary ν (we will later focus on ν = 1). The physical states of this system can be found by quantizing the corresponding matrix model (2.18) which results in the quantum Calogero model with the following Hamiltonian (we take ω = B = 1) [19,13,20] …”

Anyons and Lowest Landau Level Anyons

“…ω is small, which means, here, that the dimensionless quantity βω is small. The N -body spectrum, as given in (43), allows to compute, at leading order in βω → 0, the Z i 's for i ≤ n, and thus the b n 's…”

“…It happens that it is possible to find another N -body microscopic Hamiltonian which leads to the thermodynamics (67). Consider, in one dimension, the integrable N -body Calogero model [43] with inverse-square 2-body interactions…”