Sturm oscillation theorem

For a real one-dimensional Sturm-Liouville problem with separated self-adjoint boundary conditions and discrete simple spectrum, the eigenfunction for the $n$th eigenvalue has exactly $n$ interior zeros, numbering from zero. For a confining real Schrödinger operator on the line, the same ordering makes the ground state nodeless and the first excited state have one node. If the potential is even, uniqueness and reflection symmetry imply that these states are respectively even and odd.