On the eigenvalues of the Hamiltonian of the harmonic oscillator with the interaction (II)

We study the behaviour of eigenenergies of the operator H(λ(g).g) = H0 + λ(g)x2(1 + gx2) with H0 = −d2dx2 + x2 and λ(g). g > 0, as functions of the parameter η = g−12 near g = ∞ when λ(g) = g12, 1 = 1, 2, 3. It will be shown that, while in the first two cases the eigenvalues can be expressed as power series of η > 0, in the third case we have a divergent behaviour due to the presence of a term equal to 1η. Furthermore, apart from such a divergent term, in this case the eigenvalues approximate those of the harmonic oscillator with an attractive δ-type interaction generated by the potential 1(1 + x2) by means of a suitable scaling in η.