http://www.physicspages.com/2013/01/17/harmonic-oscillator-in-3-d-spherical-coordinates/ http://quantummechanics.ucsd.edu/ph130a/130_notes/node244.html
These are two links that have roughly the same proof of the energy levels of a 3-D harmonic oscillator, using spherical coordinates. The proofs use a step where the function is expressed as a power series. The fact that that power series counts up by integers 0,1,2.. produces the fact that the energy levels are quantized with values hω(n + l + 3/2).
Why couldn't you define the power series to count up by 0, 0.5, 1.5, 2.5, 3.5, .. or some other weird method and then come up with a result of different energy levels?
Chappers basically answered it by linking to the Frobenius method.
A.P. and Chappers further clarify why one need not worry that there are other lurking solutions that would only uncover if you expanded to think about alternate indicing representation.