  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.
