S. A. Fulling
Spectral Oscillations, Periodic Orbits, and Scaling
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ABSTRACT.  The eigenvalue density of a quantum-mechanical system exhibits 
oscillations, determined by the closed orbits of the corresponding 
classical system; this relationship is simple and strong for waves in 
billiards or on manifolds, but becomes slightly muddy for a Schrodinger 
equation with a potential, where the orbits depend on the energy. We 
discuss several variants of a way to restore the simplicity by rescaling 
the coupling constant or the size of the orbit or both. In each case 
the relation between the oscillation frequency and the period of the 
orbit is inspected critically; in many cases it is observed that a 
characteristic length of the orbit is a better indicator. When these 
matters are properly understood, the periodic-orbit theory for generic 
quantum systems recovers the clarity and simplicity that it always had 
for the wave equation in a cavity. Finally, we comment on the alleged 
"paradox" that semiclassical periodic-orbit theory is more effective 
in calculating low energy levels than high ones.