# Discrete solutions to PDEs

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!