  We investigate the use of the sinc collocation and harmonic oscillator bases
for solving a two-particle system bound by a Gaussian potential described by
the radial Schr\"odinger equation. We analyze the properties of the bound state
wave functions by investigating where the basis-state wave functions break down
and relate the breakdowns to the infrared and ultraviolet scales for both
bases. We propose a correction for the asymptotic infrared region, the long
range tails of the wave functions. We compare the calculated bound state
eigenvalues and mean square radii obtained within the two bases. From the
trends in the numerical results, we identify the advantages and disadvantages
of the two bases. We find that the sinc basis performs better in our
implementation for accurately computing both the deeply- and weakly-bound
states whereas the harmonic oscillator basis is more convenient since the
basis-state wave functions are orthogonal and maintain the same mathematical
structure in both position and momentum space. These mathematical properties of
the harmonic oscillator basis are especially advantageous in problems where one
employs both position and momentum space. The main disadvantage of the harmonic
oscillator basis as illustrated in this work is the large basis space size
required to obtain accurate results simultaneously for deeply- and weakly-bound
states. The main disadvantage of the sinc basis could be the numerical
challenges for its implementation in a many-body application.
