July 2012
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Day July 12, 2012

Spectral Singularities Do Not Correspond to Bound States in the Continuum

Ali Mostafazadeh

We show that, contrary to a claim made in arXiv:1011.0645, the von Neumann-Winger bound states that lie in the continuum of the scattering states are fundamentally different from Naimark’s spectral singularities.


Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

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

Boris F. Samsonov

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.

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

Supersymmetric η operators

Boris F. Samsonov

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.


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