Sturmian basis functions for the harmonic oscillator

We define Sturmian basis functions for the harmonic oscillator, and investi gate whether recent insights into Sturmians for Coulomb-like potentials can be extended to this important potential. We also treat many-body problems such as coupling to a bath of harmonic oscillators. Comments on coupled osc illators and time-dependent potentials are also made. It is argued that the Sturmian method amounts to a nonperturbative calculation of the energy lev els, but the limitations of the method is also pointed out, and the cause o f this limitation is found to be related to the divergence of the potential . Thus the divergent nature of the anharmonic potential leads to the Sturmi an method being less accurate than in the Coulomb case. We discuss how modi fied anharmonic oscillator potentials, which are well behaved at infinity, leads to a rapidly converging Sturmian approximation. [S1050-2947(99)10107- 0].