The Gaussian orthogonal ensemble with missing and spurious levels: A model for experimental level-spacing distributions

This paper reports analytical approximations to the level-spacing distributions for two types of eigenspectra: a Gaussian orthogonal ensemble (GOE) eigenspectrum, a fraction of whose levels are omitted, and the superposition of a fractional GOE distribution with levels from a random sequence. The combined distribution would be expected when a spectrum with a GOE spacing distribution is observed, but with insufficient dynamic range to observe all the lines, and with the possibility of contamination from other sequences or from noise identified as spectral lines. Berry recently demonstrated that GOE type spacing statistics are expected in the semiclassical limit of a classically chaotic Hamiltonian. Therefore, the present formula may prove useful in testing for quantum signatures of classical chaos in quantum molecular spectra.