A note on the eigenvalues of the Hamiltonian of the harmonic oscillator perturbed by the potential λx2(1 + gx2)

The energy of the ground state and that of the first excited level of the harmonic oscillator perturbed by the potential λx21 + gx2 (with both λ and g positive) are determined in the range of small λ and large g by means of a perturbation argument for the Fredholm determinant of the Birman-Schwinger kernel of the harmonic oscillator perturbed by the potential −λg(1 + gx2. It is seen that the eigenvalues are given by smooth functions of the two parameters λ and η = g−12 in the neighbourhood of (0,0).