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: $E^n_{\pm}(λ)$ 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 $λ=λ_{c}$. To re-emphasize, sharply at $λ=λ_{*}$ and $λ_{c}$, two real eigenvalues coincide, nevertheless their corresponding eigenfunctions become identical or linearly dependent and the Hamiltonian looses diagonalizability.