Accidental crossings of eigenvalues in the one-dimensional complex PT-symmetric Scarf-II potential

Abstract

So far, the well known two branches of real discrete spectrum of complex PT-symmetric Scarf-II potential are kept isolated. Here, we suggest that these two need to be brought together as doublets: E±n(λ) with n=0,1,2… . Then if strength (λ) of the imaginary part of the potential is varied smoothly some pairs of real eigenvalue curves can intersect and cross each other at λ=λ⁎; this is unlike one-dimensional Hermitian potentials. However, we show that the corresponding eigenstates at λ=λ⁎ are identical or linearly dependent denying degeneracy in one dimension, once again. Other pairs of eigenvalue curves coalesce to complex-conjugate pairs completing the scenario of spontaneous breaking of PT-symmetry at λ=λc. To re-emphasize, sharply at λ=λ⁎ and λc, two real eigenvalues coincide, nevertheless their corresponding eigenfunctions become identical or linearly dependent and the Hamiltonian looses diagonalizability.