On quantisation using periodic classical orbits

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Abstract
Two approaches to semiclassical quantisation of integrable systems using periodic classical orbits are considered. They both lead to approximate formulae for the density of states function (a delta function at each energy level). The first, due to Gutzwiller (1971), involves a sum over isolated stable periodic orbits of the system, and leads to the harmonic approximation to the eigenvalues. The second, due to Berry and Tabor (1976), involves a sum over families of periodic orbits, and leads to the EBK ('torus') approximation to the eigenvalues. Here, the author extracts a modified form of the Gutzwiller series from the Berry-Tabor series by using a uniform approximation, and hence show that the complete spectrum involves both these series. The analysis demonstrates that genuine semiclassical quantisation rules for generic systems, using periodic orbits, will involve uniform approximation, which more closely reflects the underlying classical structure than do the existing stationary phase approximations.

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