Generalized oscillator representations for generalized Calogero Hamiltonians

Abstract

This paper is a natural continuation of the previous paper \cite{TyuVo13} where generalized oscillator representations for Calogero Hamiltonians with potential $V(x)=\alpha /x^2$, $\alpha \ge -1/4$, were constructed. In this paper, we present generalized oscillator representations for all generalized Calogero Hamiltonians with potential $V(x)=g_1/x^2+g_2{x}^2$, $g_1\ge -1/4$, $g_2>0$. These representations are generally highly nonunique, but there exists an optimum representation for each Hamiltonian, representation that explicitly determines the ground state and the ground-state energy. For generalized Calogero Hamiltonians with coupling constants $g_1<-1/4$ or $g_2<0$, generalized oscillator representations do not exist in agreement with the fact that the respective Hamiltonians are not bounded from below.

39 pages, LaTeX-2e. arXiv admin note: substantial text overlap with arXiv:1211.6331