# Schrodinger equation subject to boundary condition implies quantization

I am in Section 2.4 of Sakurai's Modern QM, 3rd Ed. We have just deduced the time independent Schrodinger equation for energy eigenstates $$u_k(\vec x)$$ having eigenvalue $$E_k$$:

$$-\frac{\hbar^2}{2m}\nabla^2 u_k(\vec x)+V(\vec x)u_k(\vec x)=E_ku_k(\vec x)~~. \tag{1}$$

We wish to consider bound states so we will impose an appropriate boundary condition:

$$\lim\limits_{|\vec x|\to\infty} u_k(\vec x)=0~~, \tag{2}$$

with the understanding that

$$E_k<\lim\limits_{|\vec x|\to\infty} V(\vec x)~~.$$

Sakurai writes the following.

We know from the theory of PDEs that (1) subject to boundary condition (2) allows nontrivial solutions only for a discrete set of values of $$E_k$$.

I have worked through proving the discrete spectrum before, but I do not remember how it works. Can you tell me?

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.

Also, See

Discreteness of set of energy eigenvalues

Sturm-Liouville theory

Note: The answer doesn't belong to me! It originally belongs to user14717