Calogero model, deformed oscillators and the collapse

Abstract

We discuss the behavior of the Calogero model and the related model of deformed oscillators with the S_N extended Heisenberg algebra for a special value of the constant of interaction/statistical parameter nu. The problem with finite number of deformed oscillators is analyzed in the algebraic approach, while collective-field theory has been used to investigate the large-N limit. In this limit, system reduces to a large number of collapsing (free) particles, for nu=-1/N.Comment: Revtex, 7 pages, final version, to appear in PL

Aspects of Generalized Calogero Model

“…This critical point was first noticed by analysing Gram matrices of the scalar products of Fock-space states in ordinary Calogero model [11] and further confirmed by large-N collective field theory approach to the same model in Ref. [12]. For the initial Hamiltonian (4), which is not unitary equivalent toH, the conditions (17,18) demand that some ν ij < 0 and, consequently, the norm of the wave function (5) blows up at the critical point.…”

“…By applying the similarity transformation, we have obtained the simpler HamiltonianH (7), on which we have performed the Fock-space analysis and found some of its excited (collective) states (12) and their energies (13). The spectrum of collective modes is linear, equidistant and degenerate.…”

“…Similarly, the repeated action of the operator A + 2 on the vacuum |0 reproduces hypergeometric function, which reduces to associated Laguerre polynomials L ε 0 −1 n 2 +ε 0 −1 (2ωT + ) for certain values of parameters. The states (12) are eigenstates of theH with the energy eigenvalues (cf. last two equations in Eqs.…”

Calogero Model(s) and Deformed Oscillators

“…This means that the relative coordinates, the relative momenta and the relative energy are all zero at the critical value of the parameter ν. This can be also verified independently by applying large-N collective field theory to the HamiltonianH [26]. However, one should bear in mind that the interval ν ∈ (− 1 N , 0) is physically acceptable for the HamiltonianH, describing N oscillators with S N -extended Heisenberg algebra, but it is not allowed for the original Calogero Hamiltonian (1) since the wave functions, containing Jastrow factor, diverge at the coincidence point for negative values of ν.…”

“…We closely follow references [14,26,27]. In order to simplify the analysis, we extract Jastrow factor ∆ from the ground state Ψ 0 (x 1 , x 2 , .…”