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Failure of semiclassical methods to predict individual energy levels
Prosen, Tomaž, 1970- ; Robnik, Marko, 1954-
We argue that semiclassical methods quite generally cannot predict the individual energy levels not even in the semiclassical limit of small but finite h and when the number of energy levels goes to ...infinity. By this we 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 we show for the case of the integrable circular billiard and the one-dimensional potential ▫$U_0$▫/▫$cos^2$▫(▫$\alpha$▫x) by comparing the corus 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 we believe that our conslusion is generally valid. We 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 we suggest that the non-universal approach to the limiting energy level statistics is also correctly described by the semiclassical theory. We discuss the validity of the semiclassical methods in the light of our negative and positive findings. In addition we 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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