the Sturm oscillation theorem
/ STURM /
Look at the vibration modes of a string in order: the fundamental is a single smooth hump, the second mode has one still point in the middle, the third has two, and so on. The higher the mode, the more times it crosses the rest line. This counting is not a coincidence — it is a precise law. The Sturm oscillation theorem says exactly how many times each eigenfunction must cross zero, ranking the modes by their wiggliness.
For a regular Sturm-Liouville problem with eigenvalues ordered lambda_1 < lambda_2 < lambda_3 < ... and corresponding eigenfunctions y_1, y_2, y_3, ..., the theorem states that the n-th eigenfunction y_n has exactly n - 1 zeros inside the open interval (a, b). So the first eigenfunction never crosses zero in the interior, the second crosses once, the third twice, and so forth — the zero count climbs by one with each step up the ladder of eigenvalues. A companion result, the Sturm separation theorem, adds that the zeros of consecutive eigenfunctions interlace: between any two zeros of y_n lies exactly one zero of y_(n+1).
This theorem is what makes the spectrum legible. It guarantees the eigenvalues form a genuine increasing sequence with no gaps or repeats in the regular case, and it gives you a free diagnostic: count the interior zeros of an eigenfunction and you know its rank. It is the reason the energy levels of a quantum particle in a box can be labelled by how many nodes the wavefunction has, and it links directly to the comparison theorem, since more crossings is exactly the oscillation that a larger eigenvalue produces.
For y'' + lambda y = 0 on [0, pi] with zero ends, y_n = sin(n x). The first, sin(x), has no zero inside (0, pi); the second, sin(2x), has one (at x = pi/2); the third, sin(3x), has two (at pi/3 and 2pi/3). The interior-zero count is always n - 1, exactly as the theorem promises.
The n-th mode has exactly n-1 interior nodes — counting nodes tells you the rank without computing the eigenvalue.
The clean zero-count holds for the regular Sturm-Liouville problem. In singular cases, especially on infinite intervals or where the spectrum is partly continuous, the node-counting picture can break down, so do not assume 'n-th mode has n-1 nodes' for every problem you meet.