What mathematical condition can be invoked to justify quantization?
I would say that it is the boundary conditions by which the quantization is justified. Because thereby, only certain wave functions are permissible for the system, which result in zero at the boundary conditions. These wave functions possess then again also only discrete energies, whereby only these are permissible for the system and are quantized with it.
The answer lies in the experimental fact that "particles are described by waves". (Now don't ask "why particles are described by waves". Its an experimental fact and is the seed or point of nucleation of quantum mechanics.)
Like in musical instruments, when we try to confine any wave, for example be it a wave on a string on a guitar or air waves(sound waves) in a flute, all wavelengths are not allowed to build up. Only those wavelengths which respect the boundary are allowed to exist. (This happens because boundary is a place where the wave must cease to exist or should not vibrate. So the boundary has to be a node and the wave has to satisfy this condition by adjusting its wavelength suitably). Since wavelength is always related to frequency by dispersion relation, you can say that only those frequencies are allowed which respect the boundary.
Similarly when we restrict a matter wave (quantum mechanical wave representing the particle) in a potential boundary, only those selected wavelengths are allowed which respect the boundary. By dispersion relation, we can find out the frequencies for these special wavelengths. Then you know that frequency is the representative of energy. Thus energy is quantized.
Mathematical solutions are not different from physical states of a particle. Putting boundary condition on differential equation means in reality we are thinking of the "potential boundaries" of the particle. Change the boundary conditions and you get different set of solutions with different set of quantized energy. The differential equation we are solving has all the information of the particle that we are considering. The mathematics that you are talking about is just describing the physics; not dictating it.
