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arXiv:1901.02357 · Journal of Mathematical Physics

New scattering features of quaternionic point interaction in non-Hermitian quantum mechanics

Mohammad Hasan, Bhabani Prasad Mandal

Published Mar 1, 2020• 31 citations
arXivHTMLPDF

Abstract

Spectral singularities have been extensively studied over the last one and half decades for different non-Hermitian potentials in non-Hermitian quantum mechanics. The nature of spectral singularities has not been studied for the case of quaternionic potential. In the present work, we perform an analytical study on scattering from a quaternionic point interaction represented by a delta function. New features of spectral singularities are observed, which are different than the case of a complex (non-quaternionic) point interaction. The most notable difference is the occurrence of spectral singularity from the lossy point interaction, which is forbidden in the case of standard non-Hermitian quantum mechanics.

DOI
10.1063/1.5117873

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Spectral singularities have been extensively studied over the last one and half decades for different non-Hermitian potentials in non-Hermitian quantum mechanics. The nature of spectral singularities has not been studied for the case of quaternionic potential. In the present work, we perform an analytical study on scattering from a quaternionic point interaction represented by a delta function. New features of spectral singularities are observed, which are different than the case of a complex (non-quaternionic) point interaction. The most notable difference is the occurrence of spectral singularity from the lossy point interaction, which is forbidden in the case of standard non-Hermitian quantum mechanics.
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 iViViV, V∈R+V \in R^{+}V∈R+. We show that these loci are elliptical and self-similar to each other in E−VE-VE−V plane where EEE is energy of the wave. The orientations of these ellipses for reflection (RRR) and transmission (TTT) amplitude around a given SS are same. This is surprising due to the different mathematical forms of RRR and TTT. In the vicinity of SS, R≈TR \approx TR≈T.
A curious feature of complex scattering potentials v(x) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., − d2 dx 2 +v(x), and its adjoint. We establish the equivalence of this description with the mathematicians ’ definition of spectral singularities for the potential v(x) = z−δ(x + a) + z+δ(x − a), where z ± and a are respectively complex and real parameters and δ(x) is the Dirac delta-function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z ± where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing PT-symmetry. PACS numbers: 03.65.-w
A curious feature of complex scattering potentials v ( x ) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., , and its adjoint. We establish the equivalence of this description with the mathematicians' definition of spectral singularities for the potential v ( x ) = z − δ( x + a ) + z + δ( x − a ), where z ± and a are respectively complex and real parameters and δ( x ) is the Dirac delta function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z ± where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing -symmetry.
This paper explores the novel concept of topological eigenvalues within the realm of non-Hermitian quantum mechanics, focusing on their role in unveiling spectral singularities. Unlike conventional Hermitian systems, non-Hermitian Hamiltonians, which describe open systems with gain and loss, exhibit complex eigenvalues and unique spectral features such as exceptional points (EPs) where eigenvalues and their corresponding eigenvectors coalesce. We introduce the idea that certain eigenvalues in non-Hermitian systems possess an inherent topological character, dictated by the geometry of the parameter space and the winding properties of the characteristic polynomial. These topological eigenvalues are shown to be intimately linked to spectral singularities, which are non-Hermitian generalizations of bound states in the continuum or points of perfect reflection/transmission, characterized by infinite scattering amplitudes. We develop a theoretical framework to identify and characterize these topological eigenvalues, demonstrating how they emerge from the interplay of non-Hermiticity and underlying symmetries. Our findings reveal that the robust nature of these topological eigenvalues provides a new lens through which to understand the exotic phenomena observed in non-Hermitian systems, including the non-Hermitian skin effect and the breakdown of traditional bulk-boundary correspondence. This work opens new avenues for designing robust non-Hermitian devices and understanding fundamental aspects of open quantum systems.
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.
The Dirac delta function potential is considered within the real Hilbert space approach for complex wave functions, as well as quaternionic wave functions. As has been previously determined, the real Hilbert space approach enables the possibility of self-interacting physical systems. The self-interaction precludes confining states, and also imposes non-stationary quantum states, both of them representing novel situations that cannot be observed in terms of quantum wave functions. These results remark the differences between quaternionic quantum mechanics ($\mathbbm H$QM) and complex quantum mechanics ($\mathbbm C$QM), and also establish a method of solving the wave equation that may be applied to a variety of different cases.