In Pauling and Wilson, Introduction to Quantum Mechanics, they offer the following intuitive reason for the discrete spectrum of a potential which is unbounded at $\pm \infty$:
This is interesting but I'm not sure I "buy" it in it's current form. Is there a way to make this argument mathematically precise?
You can think about "solving X equations in Y unknowns". When $X>Y$, you generally expect infinite solutions, when $X=Y$ you generally expect a unique solution, when $X<Y$ you generally expect no solutions. This kind of statement is not always mathematically rigorous but you can usually argue it rigorously in specific circumstances.
Pick an energy $W$ and an arbitrary point $x$. You can pick any two real values for $\psi(x)$ and $\psi'(x)$: You have two continuous adjustable parameters here. Oops, not really. Because the differential equation is homogeneous, you can assume (without loss of generality) that $\psi(x)$ is either 1 or 0. So you actually have only one continuous adjustable parameter. Let's assume from now on that $\psi(x)=1$, because the $\psi(x)=0$ case is actually equivalent to a special case of $\psi(x)=1$ with the slope going to $\pm \infty$.
"Going to zero on the right" is a constraint, so we have one constraint and one adjustable parameter. We generally expect a unique solution. In fact, I think you can rigorously prove that there is a unique solution here, because the asymptotic behavior is related monotonically to $\psi'(x)$.
Similarly, "going to zero on the left" is a constraint, so there is a unique value of $\psi'(x)$ that satisfies this constraint.
Now, we define a function $R(W)$ which is the value of $\psi'(x)$ that makes the solution go to zero on the right at energy $W$; and similarly $L(W)$ on the left. Solutions occur at crossings where $L(W) = R(W)$.
Usually when you draw two 1D curves, they cross each other only at discrete points, rather than, say, perfectly overlapping over a continuous interval.
Can we rigorously prove that you don't have the weird circumstance where $L(W) = R(W)$ over a whole continuous interval of different $W$? Presumably you can prove it somehow, but I'm not sure of the details...
The first thing I would try is to find an expression for the derivatives $L'(W)$ and $R'(W)$, in terms of $V$, and hopefully I would be able to prove that they cannot be equal over a whole interval. Something like that...
Yes, there is actually a rigorous proof, for detail's read Simon and Rid, methods of modern mathematical physics, it's the theorem 13.16. (13 roman). It holds for bounded from below hamiltonians (basically, when you can't measure infinitly low energy, that is totally obvious for a physicist, but not for a expert in spectral analysis) and if your potential function is bounded itself. When, this is a strict theorem. Actually, it's cool to read into the basis of it's proof - minimax theorem, which tells you hell'a lot about spectrum if general case.
