Failure of semiclassical methods to predict individual energy levels

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Abstract
The authors argue that semiclassical methods quite generally cannot predict the individual energy levels not even in the semiclassical limit of small but finite and when the number of energy levels goes to infinity. By this they mean that the average relative error of the semiclassical eigenvalues in units of the mean level spacing typically increases indefinitely as the energy goes to infinity or is at least bounded from below. This they show for the case of the integrable circular billiard and the one-dimensional potential Uo/cos2( alpha x) by comparing the torus quantized semiclassical eigenenergies with the exact results. Since all the various semiclassical methods such as Gutzwiller's and Bogomolny's are reduced to the torus quantization in integrable cases they believe that their conclusion is generally valid. They have theoretical arguments and strong numerical evidence (for the case of the circular billiard) that nevertheless the statistical properties of the exact energy spectra are correctly reproduced by the semiclassical approximations. It is numerically found that the energy level spacing distribution and the spectral rigidity for the exact spectrum and for the semiclassical spectrum are in excellent agreement even for finite spectra where they both deviate from the limiting Poissonian behaviour, so they suggest that the nonuniversal approach to the limiting energy level statistics is also correctly described by the semiclassical theory. They discuss the validity of semiclassical methods in the light of their negative and positive findings. In addition they find the surprising result for the previously mentioned special cases that the error distribution of the semiclassical approximation is stationary, i.e. it is independent of the energy.

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