Scattering cross section resonance originating from a spectral singularity

Abstract

Using techniques of supersymmetric quantum mechanics, scattering properties of Hermitian Hamiltonians, which are related to non-Hermitian ones by similarity transformations, are studied. We have found that the scattering matrix of the Hermitian Hamiltonian coincides with the phase factor of the non-unitary scattering matrix of the non-Hermitian Hamiltonian. The possible presence of a spectral singularity in a non-Hermitian Hamiltonian translates into a pronounced resonance in the scattering cross section of its Hermitian counterpart. This opens a way for detecting spectral singularities in scattering experiments; although a singular point is inaccessible for the Hermitian Hamiltonian, the Hamiltonian "feels" the presence of the singularity if it is "close enough". We also show that cross sections of the non-Hermitian Hamiltonian do not exhibit any resonance behavior and explain the resonance behavior of the Hermitian Hamiltonian cross section by the fact that corresponding scattering matrix, up to a background scattering matrix, is a square root of the Breit-Wigner scattering matrix.

Minor modifications are made following recommendations of Referee

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• Optical Spectral Singularities and Coherent Perfect Absorption in a Two-Layer Spherical Medium
• Nonlinear Spectral Singularities and Lasing Threshold Condition
• Spectral Singularities and Whispering Gallery Modes of a Cylindrical Gain Medium
• Generalized Unitarity and Reciprocity Relations for PT-symmetric Scattering Potentials
• Lasing Threshold Condition for Oblique TE and TM Modes, Spectral Singularities, and Coherent Perfect Absorption
• Spectral Singularities in the Surface Modes of a Spherical Gain Medium
• Perturbative Analysis of Spectral Singularities and Their Optical Realizations
• Talbot self-imaging in $\mathcal{P}\mathcal{T}$-symmetric complex crystals
• Perturbative Unidirectional Invisibility
• Hermitian Hamiltonian equivalent to a given non-Hermitian one. Manifestation of spectral singularity
• Non-Hermitian quantum rings
• Adiabatic Series Expansion and Higher-Order Semiclassical Approximations in Scattering Theory
• Supersymmetric $\eta$ operators