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

TL;DR

This paper investigates eigenvalue crossings in a one-dimensional PT-symmetric Scarf-II potential, revealing that real eigenvalues can intersect without degeneracy, and that eigenfunctions become linearly dependent at crossing points, indicating loss of diagonalizability.

Contribution

It demonstrates that eigenvalue crossings occur in the PT-symmetric Scarf-II potential with eigenfunctions becoming linearly dependent, challenging typical degeneracy expectations in one dimension.

Findings

Eigenvalue crossings occur at specific parameter values.

Eigenfunctions at crossing points are linearly dependent.

Loss of diagonalizability at crossing points.

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.