  The transformation of a classical system into its quantum counterpart is
usually done through the well known procedure of canonical quantization.
However, on non-Cartesian domains, or on bounded Cartesian domains, this
procedure can be plagued with theoretical inconsistencies. An alternative
approach is {\it affine quantization} (AQ) Fantoni and Klauder
(arXiv:2109.13447,Phys. Rev. D {\bf 103}, 076013 (2021)), resulting in
different conjugate variables that lead to a more consistent quantization
formalism. To highlight these issues, we examine a deceptively simple, but
important, problem: that of the harmonic oscillator potential on the
semibounded domain: ${\cal D} = (-b,\infty)$. The AQ version of this
corresponds to the (rescaled) system, ${\cal H} = \frac{1}{2}\Big(-\partial_x^2
+ \frac{3}{4(x+b)^2} + x^2\Big)$. We solve this system numerically for $b > 0$.
The case $b = 0$ corresponds to an {\it exactly solvable} potential, for which
the eigenenergies can be determined exactly (through non-wavefunction dependent
methods), confirming the results of Gouba ( arXiv:2005.08696 ,J. High Energy
Phys., Gravitation Cosmol. {\bf 7}, 352-365 (2021)). We investigate the limit
$b \rightarrow \infty$, confirming that the full harmonic oscillator problem is
recovered. The adopted computational methods are in keeping with the underlying
theoretical framework of AQ. Specifically, one method is an affine map
invariant variational procedure, made possible through a moment problem
quantization reformulation. The other method focuses on boundedness (i.e.
$L^2$) as an explicit quantization criteria. Both methods lead to converging
bounds to the discrete state energies; and thus confirming the accuracy of our
results, particularly as applied to a singular potential problem.
