One of the simplest non-Hermitian Hamiltonians first proposed by Schwartz (1960 Commun. Pure Appl. Math. 13 609) which may possess a spectral singularity is analyzed from the point of view of non-Hermitian generalization of quantum mechanics. It is shown that \(\eta\) operator, being a second order differential operator, has supersymmetric structure. Asymptotic behavior of eigenfunctions of a Hermitian Hamiltonian equivalent to the given non-Hermitian one is found. As a result the corresponding scattering matrix and cross section are given explicitly. It is demonstrated that the possible presence of the spectral singularity in the spectrum of the non-Hermitian Hamiltonian may be detected as a resonance in the scattering cross section of its Hermitian counterpart. Nevertheless, just at the singular point the equivalent Hermitian Hamiltonian becomes undetermined.

http://arxiv.org/abs/1207.2525
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Being chosen as a differential operator of a special form, metric \(\eta\) operator becomes unitary equivalent to a one-dimensional Hermitian Hamiltonian with a natural supersymmetric structure. We show that fixing the superpartner of this Hamiltonian permits to determine both the metric operator and corresponding non-Hermitian Hamiltonian. Moreover, under an additional restriction on the non-Hermitian Hamiltonian, it becomes a superpartner of another Hermitian Hamiltonian.

http://arxiv.org/abs/1207.2522

Mathematical Physics (math-ph); Quantum Physics (quant-ph)