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arXiv:1207.2525 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Hermitian Hamiltonian equivalent to a given non-Hermitian one: manifestation of spectral singularity

B. F. Samsonov

Published Mar 18, 2013• 14 citations
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Abstract

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.

DOI
10.1098/rsta.2012.0044

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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.
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 the corresponding scattering matrix, up to a background scattering matrix, is a square root of the Breit–Wigner scattering matrix.
Being chosen as a differential operator of a special form, the metric η operator has become unitary equivalent to a one-dimensional Hermitian Hamiltonian with a natural supersymmetric structure. We show that fixing the superpartner of this Hamiltonian permits us to determine both the metric operator and the corresponding non-Hermitian Hamiltonian. Moreover, under an additional restriction on the non-Hermitian Hamiltonian, it has become a superpartner of another Hermitian Hamiltonian. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
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.
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.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.