It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry.

More precisely, the operator

$$-\frac{d^2}{dx^2}+x^2$$

can be decomposed as

$$-\frac{d^2}{dx^2}+x^2 = \left(-\frac{d}{dx}-x\right)\left(\frac{d}{dx}-x\right)=:a^*a.$$

When I tried to see whether the same holds true in spherical coordinates I noticed a complete suprise

$$\left(-\frac{d}{dr}-\frac{n-1}{r}-r\right)\left(\frac{d}{dr}-r\right) = -\frac{d^2}{dr^2} -\frac{n-1}{r} \frac{d}{dr} +r^2+(n-1).$$

The first three terms comprise the harmonic oscillator in spherical coordinates that is

$$-\Delta_r + r^2.$$

For convenience, I discarded the angular part in this calculation.

But for some reasons I seem to get this superficial term $(n-1)$. Why is that? And can I do anything smarter to avoid this term?