Asymptotic behavior of eigen energies of non-Hermitian cubic polynomial systems

The asymptotic behavior of the eigenvalues of a non-Hermitian cubic polynomial system H = (P-2/2) + mu x(3) + ax(2) + bx, where mu, a, and b are constant parameters that can be either real or complex, is studied by extending the asymptotic energy expansion method, which has been developed for even degree polynomial systems. Both the complex and the real eigenvalues of the above system are obtained using the asymptotic energy expansion. Quantum eigen energies obtained by the above method are found to be in excellent agreement with the exact eigenvalues. Using the asymptotic energy expansion, analytic expressions for both level spacing distribution and the density of states are derived for the above cubic system. When mu = i, a is real, and b is pure imaginary, it was found that asymptotic energy level spacing increases with the coupling strength a for positive a while it decreases for negative a.