Accidental crossings of eigenvalues in 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: $E^n_{\pm}(\lambda)$ with $n=0,1,2...$. Then if strength $(\lambda)$ of the imaginary part of the potential is varied smoothly some pairs of real eigenvalue curves can intersect and cross each other at $\lambda=\lambda_{}$; this is unlike one dimensional Hermitian potentials. However, we show that the corresponding eigenstates at $\lambda=\lambda_{}$ 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 $\lambda=\lambda_{c}$. To re-emphasize, sharply at $\lambda=\lambda_{*}$ and $\lambda_{c}$, two real eigenvalues coincide, nevertheless their corresponding eigenfunctions become identical or linearly dependent and the Hamiltonian looses diagonalizability.