Spectral Oscillations, Periodic Orbits, and Scaling

Details Spectral Oscillations, Periodic Orbits, and Scaling Spectral Oscillations, Periodic Orbits, and Scaling

2000-12-14

arxiv.org/abs/quant-ph/0012070v3

J. Phys. A 35, 4049-4066 (2002)

Physics

Quantum Physics

19 pages, RevTeX4 with PicTeX. Minor improvements in content, new references, typos corrected

Scientific paper

10.1088/0305-4470/35/18/305

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.

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