TY - JOUR
T1 - Hermitian Hamiltonian equivalent to a given non-Hermitian one: manifestation of spectral singularity
JF - Philosophical Transactions of the Royal Society A: Mathematical,
Physical and Engineering Sciences
JO - Philos Transact A Math Phys Eng Sci
M3 - 10.1098/rsta.2012.0044
VL - 371
IS - 1989
AU - Samsonov, Boris F.
Y1 - 2013/04/28
UR - http://rsta.royalsocietypublishing.org/content/371/1989/20120044.abstract
N2 - One of the simplest non-Hermitian Hamiltonians, first proposed by Schwartz in 1960, that may possess a spectral singularity is analysed from the point of view of the non-Hermitian generalization of quantum mechanics. It is shown that the η operator, being a second-order differential operator, has supersymmetric structure. Asymptotic behaviour of the 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 a 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.
ER -