Oscillations in eigenvalue density are associated with closed orbits of the corresponding classical (or geometrical optics) system. Although invisible in the heat-kernel expansion, these features determine the nonlocal parts of propagators, including the Casimir energy. I review some classic work, discuss the connection with vacuum energy, and show that when the coupling constant is varied with the energy, the periodic-orbit theory for generic quantum systems regains the clarity and simplicity that it always had for the wave equation in a cavity.
