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 ₂ x ² , g ₁ ≥ ?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.