The interplay of supersymmetry and PT symmetry in quantum mechanics: a case study for the Scarf II potential

Motivated by the duality of normalizable states and the presence of the quasi-parity quantum number q = ± 1 in symmetric (non-Hermitian) quantum mechanical potential models, the relation of symmetry and supersymmetry (SUSY) is studied. As an illustrative example, the invariant version of the Scarf II potential is presented, and it is shown that the 'bosonic' Hamiltonian has two different 'fermionic' SUSY partner Hamiltonians (potentials) generated from the ground-state solutions with q = 1 and q = −1. It is shown that the 'fermionic' potentials cease to be invariant when the symmetry of the 'bosonic' potential is spontaneously broken. A modified symmetry inspired SUSY construction is also discussed, in which the SUSY charge operators contain the antilinear operator . It is shown that in this scheme the 'fermionic' Hamiltonians are just the complex conjugates of the original 'fermionic' Hamiltonians, and thus possess the same energy eigenvalues.

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