Supersymmetric quasi-Hermitian Hamiltonians with point interactions on a loop

We explore some aspects of {\mathcal{PT}} -symmetric Hamiltonians with two point interactions. We determine classes of point interactions for which the Hamiltonians are supersymmetric. We prove that these Hamiltonians are quasi-Hermitian and find a very simple formula for the metric operator Θ and its square root rhov as well. Further, we present the quasi-Hermitian Hamiltonian (with one-point interaction) with a continuous spectrum.

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