Stochastic formulation of energy-level statistics

It is shown that the joint distribution of energy eigenvalues for systems with a varying degree of nonintegrability which has been obtained dynamically by T. Yukawa [Phys. Rev. Lett. 54, 1883 (1985)] can also be deduced by putting his equations of motion in the form of stochastic differential equations. We obtain an interpolation formula for the nearest-neighbor-spacing distribution as a smooth one-parameter family of density functions $P_{\lambda }$(S), 0≤λ<∞. This distribution retains a nonanalytic nature near λ→0; when λ=0 it agrees with the Poissonian distribution but whenever λ≠0 it is proportional to S for small S, as predicted by M. Robnik [J. Phys. A 20, L495 (1987)]. A considerable improvement on the agreement between the energy-level histogram in a real system (hydrogen in a magnetic field) and theoretical formulas which have been studied by Wintgen and Friedrich [Phys. Rev. A 35, 1464 (1987)] is demonstrated.