A note on quantum chaology and gamma approximations to eigenvalue spacings for infinite random matrices

Quantum counterparts of certain simple classical systems can exhibit chaotic behaviour through the statistics of their energy levels. Gamma distributions do not precisely model the various analytic systems discussed here, but some features may be useful in studies of qualitative generic properties in applications to data from real systems which manifestly seem to exhibit behaviour reminiscent of near-random processes. We use known bounds on the distribution function for eigenvalue spacings for the Gaussian orthogonal ensemble (GOE) and show that gamma distributions, which have an important uniqueness property, can yield an approximation similarly good, except near the origin, to that of the widely used Wigner surmise. This has the advantage that then both the chaotic and non chaotic cases fit in the information geometric framework of the manifold of gamma distributions, which has been the subject of recent work on neighbourhoods of randomness for more general stochastic systems