The spectral singularity (SS) from a non-Hermitian potential is one of the most remarkable scattering feature of non-Hermitian quantum mechanics. At the spectral singular point, the scattering amplitudes diverge to infinite. This phenomena have been extensively studied over the last two decades. The previous studies have suggested the need of extremely fine control of the various system parameters for practical applications of SS. However no study have been carried out to understand the behavior of the scattering amplitude in the vicinity of SS. Such study would be important towards the practical application of SS where it is desired to maintain outgoing scattering amplitude to a specific value. The behavior of the loci of constant scattering amplitude in the neighborhood of SS are studied for a pure imaginary barrier potential $iV$ , $V \in R^{+}$ . We show that these loci are elliptical and self-similar to each other in $E-V$ plane where $E$ is energy of the wave. The orientations of these ellipses for reflection ( $R$ ) and transmission ( $T$ ) amplitude around a given SS are same. This is surprising due to the different mathematical forms of $R$ and $T$ . In the vicinity of SS, $R \approx T$ .The spectral singularity (SS) from a non-Hermitian potential is one of the most remarkable scattering feature of non-Hermitian quantum mechanics. At the spectral singular point, the scattering amplitudes diverge to infinite. This phenomena have been extensively studied over the last two decades. The previous studies have suggested the need of extremely fine control of the various system parameters for practical applications of SS. However no study have been carried out to understand the behavior of the scattering amplitude in the vicinity of SS. Such study would be important towards the practical application of SS where it is desired to maintain outgoing scattering amplitude to a specific value. The behavior of the loci of constant scattering amplitude in the neighborhood of SS are studied for a pure imaginary barrier potential $iV$, $V \in R^{+}$. We show that these loci are elliptical and self-similar to each other in $E-V$ plane where $E$ is energy of the wave. The orientations of these ellipses for reflection ($R$) and transmission ($T$) amplitude around a given SS are same. This is surprising due to the different mathematical forms of $R$ and $T$. In the vicinity of SS, $R \approx T$.

arXiv:1910.05138Oct 10, 20190 citations

Formation of Singularities

Yves Pomeau A, Minh-Binh Tran

Lecture notes in physicsJan 1, 20190 citations

Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at Real Energies

Alí Mostafazadeh

Spectral singularities are spectral points that spoil the completeness of the eigenfunctions of certain non-Hermitian Hamiltonian operators. We identify spectral singularities of complex scattering potentials with the real energies at which the reflection and transmission coefficients tend to infinity, i.e., they correspond to resonances having a zero width. We show that a waveguide modeled using such a potential operates like a resonator at the frequencies of spectral singularities. As a concrete example, we explore the spectral singularities of an imaginary PT-symmetric barrier potential and demonstrate the above resonance phenomenon for a certain electromagnetic waveguide.Spectral singularities are spectral points that spoil the completeness of the eigenfunctions of certain non-Hermitian Hamiltonian operators. We identify spectral singularities of complex scattering potentials with the real energies at which the reflection and transmission coefficients tend to infinity, i.e., they correspond to resonances having a zero width. We show that a waveguide modeled using such a potential operates like a resonator at the frequencies of spectral singularities. As a concrete example, we explore the spectral singularities of an imaginary PT-symmetric barrier potential and demonstrate the above resonance phenomenon for a certain electromagnetic waveguide.

arXiv:0901.4472Physical Review LettersJun 5, 2009493 citations

Spectral functions and Green’s functions: A critique

A. K. Mishra, S.K. Rangarajan

Proceedings of the Indian Academy of Sciences - Section AOct 1, 19863 citations

The Problem of Confluent Singularities

Bernie G. Nickel

Springer eBooksJan 1, 198211 citations

iV

V \in R^{+}

. We show that these loci are elliptical and self-similar to each other in

E-V

plane where

E

is energy of the wave. The orientations of these ellipses for reflection (

R

) and transmission (

T

) amplitude around a given SS are same. This is surprising due to the different mathematical forms of

R

and

T

. In the vicinity of SS,

R \approx T