 00501 S. A. Fulling
 Spectral Oscillations, Periodic Orbits, and Scaling
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Dec 15, 00

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Abstract. The eigenvalue density of a quantummechanical 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 periodicorbit 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 periodicorbit theory is more effective
in calculating low energy levels than high ones.
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