With no repulsive Coulomb interaction you should get a result which corresponds to the relative energy of a non-interacting system. Make sure your results are stable as functions of \( \rho_{\mathrm{max}} \) and the number of steps.

We are only interested in the ground state with \( l=0 \). We omit the center-of-mass energy.

For specific oscillator frequencies, the above equation has analytic answers, see the article by M. Taut, Phys. Rev. A 48, 3561 - 3566 (1993). The article can be retrieved from the following web address http://prola.aps.org/abstract/PRA/v48/i5/p3561_1.