Large-order perturbation expansions for the charged harmonic oscillator

The charged oscillator defined by the Hamiltonian H=-d 2 /dr 2 +r 2 + lambda /r in the domain (0, infinity ), is a particular case of the family of spike oscillators, which does not behave as a supersingular Hamiltonian. This problem is analysed around the three regions lambda to infinity , lambda to 0 and lambda to - infinity by using Rayleigh-Ritz large-order perturbative expansions. A path is found to connect the large lambda regions with the small lambda region by means of the renormalization of the series expansions in lambda . Finally, the Riccati-Pade method is used to construct an implicit expansion around lambda to 0 which extends to very large values of mod lambda mod .