Resonant periodic orbits and the semiclassical energy spectrum

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Abstract
The semiclassical density of states depends, according to the periodic-orbit sum formula, on the linear stability of the orbits. This means, however, that contributions from the marginally stable or 'resonant' orbits, which necessarily accompany stable ones, diverge unphysically. The remedy for a system of two degrees of freedom is found to lie in the classical non-linear normal forms for periodic orbits, which describe how satellite periodic orbits coalesce with the central one as resonance is approached ( in to 0). Through these forms the resonant contributions are expressed as diffraction integrals (the first few being 'diffraction catastrophes') uniformly valid in in and h(cross), and finite even for in to 0 provided h(cross)=0. An extension is proposed to incorporate, jointly, multiple resonances found in repetitions of orbits.

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