Level Spacing Distributions and Quantum Chaos in Hermitian and non-HermitianSystems

Abstract

The correspondence between quantum level spacing distributions and classicalmotion of 1-D PT symmetric non-Hermitian systems is investigated using two PT symmetric complex potentials: complex rational power potentialV1(x) = (ix)(2n + 1)/m and general polynomial potential V2(x) =x2M + ib1 x2M − 1 + b2 x2M − 2 + ... + ib2M − 1x. The levelspacing distribution of V1 has two forms. When 2n + 1 − 2m ispositive, the level spacing distribution of real eigen values assumes adecreasing power function, while it behaves as an increasing power functionwhen 2n + 1 − 2m is negative. The PT symmetry of this system isspontaneously broken as 2n + 1 − 2m becomes negative. This changemanifests itself in classical mechanics as it is found by Bender et al. However, it was found that the change in the form of level spacingdistribution mentioned above is not due to the spontaneous breaking down of PT symmetry. Level spacing distribution of V2 assumes an increasing powerfunction when order of the polynomial is greater than two.