Hermitian Hamiltonian equivalent to a given non-Hermitian one. Manifestation of spectral singularity

One of the simplest non-Hermitian Hamiltonians first proposed by Schwartz (1960 {it Commun. Pure Appl. Math.} tb{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.