I am reading Sakurai's Modern Quantum Mechanics. He notes on the solutions to the time-independent Schrodinger equation that
We know from the theory of partial differential equations that [the time independent Schrodinger equation in 3D] subject to the boundary conditions (2.4.13) (i.e. solution $u(\textbf{r})\rightarrow 0$ as $\textbf{r}\rightarrow 0$) allows non-trivial solutions only for discrete set of values of the energy E.
It has puzzled me for a long time a to why some equations permit only a discrete set, whilst others a continuous, set of solutions. I have been trying to find a proof of this, or at least some discussion as to why it ould be the case for the Schrodinger equation and the above conditions (I suspect there won't be a general argument for PDEs in general) . However putting the words 'discrete' and 'PDE' together in a Google search seems only to yield pages on numerical solutions to PDEs. I have also tried to look more generally at the 'theory of PDEs' with no luck. I would appreciate if someone could provide a link on the topic!
The answer to this question was not resolved until roughly 25 years after the Schrodinger equation was written down. See Tosio Kato's work "Fundamental Properties of Hamiltonian Operators of Schrodinger Type".
In the paper, Kato shows that for the potentials that we care about (i.e., Coulombic), the spectrum is discrete. As you read through Kato's paper, you will begin to understand why all physics texts attempt to sweep this issue under the rug: It requires very advanced functional analysis to prove.