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

• Mixing of two branches of discrete eigenvalues of PT-symmetric Scarf II.
• Accidental crossings of energy eigenvalues in one dimension, yet no degeneracy.
• Loss of diagonalizability of Hamiltonian at the points of level crossings.
• Negative energy poles of transmission coefficients.
• A rare property of Jacobi polynomials used and proved.

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…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⁎λ=λ⁎; this is unlike one-dimensional Hermitian potentials. However, we show that the corresponding eigenstates at λ=λ⁎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=λcλ=λc. To re-emphasize, sharply at λ=λ⁎λ=λ⁎ and λcλc, two real eigenvalues coincide, nevertheless their corresponding eigenfunctions become identical or linearly dependent and the Hamiltonian looses diagonalizability.