real discrete eigenvalues
If you tap a guitar string, it does not ring at every conceivable frequency — it picks out a discrete ladder of tones. That selectivity is the physical face of a mathematical fact: a regular Sturm-Liouville problem has eigenvalues that are real, and they form a discrete, increasing sequence with no upper bound. The allowed lambda values are isolated, countable, and ordered: lambda_1 < lambda_2 < lambda_3 < ... tending to infinity.
Reality of the eigenvalues follows from self-adjointness, exactly as for a symmetric matrix. Suppose L[y] = -lambda w y with y complex; multiply by the complex conjugate, integrate, integrate by parts using the self-adjoint structure and the boundary conditions, and you find lambda must equal its own conjugate, so it is real. The discreteness and the unbounded increase come from oscillation theory: the n-th eigenfunction has exactly n - 1 interior zeros, and as lambda climbs, solutions oscillate faster, so eigenfunctions and their eigenvalues come one at a time rather than as a continuum. There is also a lowest eigenvalue lambda_1 — a ground state — but no highest.
These three properties are the load-bearing assumptions behind every eigenfunction expansion. Discreteness is why you sum over n rather than integrate over a continuum (contrast the Fourier transform, whose spectrum is continuous). Reality is why the modes are honest oscillations rather than growing or decaying complex things. The ordered ladder with a smallest eigenvalue is what lets variational methods bracket lambda_1 from above, and why the lowest mode dominates long-time heat decay or the fundamental tone of an instrument.
For y'' + lambda y = 0 with y(0) = y(L) = 0 the eigenvalues are lambda_n = (n pi / L)^2 for n = 1, 2, 3, ...: all real, all positive, spaced ever further apart, marching to infinity — and the n-th eigenfunction sin(n pi x / L) has exactly n - 1 interior zeros.
The eigenvalue ladder is the set of allowed squared frequencies of a clamped string; the zero-counting rule labels the modes.
Discreteness is guaranteed only for regular problems on a finite interval (and some singular ones); on an infinite domain a Sturm-Liouville operator can have a continuous spectrum instead, which is why the Fourier transform integrates rather than sums.