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

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