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: $E_{\pm }^n(\lambda )$ with $n=0,1,\mathrm{2...}$. 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 $\lambda ={\lambda }_{\ast }$; this is unlike one dimensional Hermitian potentials. However, we show that the corresponding eigenstates at $\lambda ={\lambda }_{\ast }$ 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 $\lambda ={\lambda }_{\ast }$ and ${\lambda }_c$, two real eigenvalues coincide, nevertheless their corresponding eigenfunctions become identical or linearly dependent and the Hamiltonian looses diagonalizability.

16 pages, 4 figures and two Tables

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Cited by in corpus (7)

• Two patterns of PT-symmetry breakdown in a non-numerical six-state simulation
• Analytical solutions for the radial Scarf II potential
• Bosonic pair creation and the Schiff-Snyder-Weinberg effect
• Unusual isospectral factorizations of shape invariant Hamiltonians with Scarf II potential
• Supersymmetric approach to exact solutions of $(1+1)$-dimensional time-independent Klein-Gordon equation : Application to a position-dependent mass and a $\mathcal{P}\mathcal{T}$-symmetric vector potential
• Dirichlet spectrum of the paradigm model of complex PT-symmetric potential: $V(x)=-(ix{)}^N$
• Real discrete spectrum of complex PT-symmetric scattering potentials