Generalized oscillator representations for generalized Calogero Hamiltonians

We construct generalized oscillator representations for all generalized Calogero Hamiltonians with the potential $V(x)=g_1/x^2+g_2x^2$, $g_1\ge-1/4$, $g_2>0$. These representations are generically nonunique, but for each Hamiltonian, there exists an optimum representation explicitly determining the ground state and its energy. For generalized Calogero Hamiltonians with coupling constants $g_1<-1/4$ or $g_2<0$, generalized oscillator representations do not exist, which agrees with the fact that the corresponding Hamiltonians are not bounded from below.