S. Banerjee, R. Roychoudhury
Using purely physical arguments it is claimed that for ID Schrodinger operators with complex PT- Symmatric potentials having a purely real attractive potential well and a purely imaginary repulsive part,bound state eigenvalues will be discrete and real.This has been illustrated with several potentials possessing similar properties.
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We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operator and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-$x$ plane (Rep. Prog. Phys. {\bf 70} (2007) 947).
We analyze recent results on a Dirac oscillator. We show that the truncation of the Frobenius series does not yield all the eigenvalues and eigenfunctions of the radial equation. For this reason the eigenvalues reported by the authors are useless and the prediction of allowed oscillator frequencies meaningless.
We show that the author of a recent paper [arXiv:2008.04929] put forward some false statements about the eigenstates of Hermitian and non-Hermitian systems. We conjecture that one of the non-Hermitian Hamiltonians for a one-dimensional lattice is similar to an Hermitian one and, consequently, exhibits real eigenvalues. Present theoretical analysis of the eigenvalue equation suggests that one of the sets of numerical results in the criticized paper may not be correct.
We analyze recent results for a harmonic oscillator in an environment with a pointlike defect. We show that the allowed oscillator frequencies predicted by the authors stem from a misinterpretation of the exact solutions of a conditionally solvable eigenvalue equation. Also the exact eigenvalues derived by those authors are meaningless because they belong to different quantum-mechanical models.
We analyze the results of a paper on "the arising of a Rashba-like coupling, a Zeeman-like term and a Darwin-like term induced by Lorentz symmetry breaking effects in the non-relativistic quantum dynamics of a spin-1/2 neutral particle interacting with external fields". We show that the authors did not obtain the spectrum of the eigenvalue equation but only one eigenvalue for a specific relationship between model parameters. In particular, the existence of allowed cyclotron frequencies conjectured by the authors is a mere artifact of the truncation condition used to obtain exact solutions to t…
In this comment we show that the eigenvalues of a quartic anharmonic oscillator obtained recently by means of the asymptotic iteration method may not be as accurate as the authors claim them to be.
In the paper "Observation of Fast Evolution in Parity-Time-Symmetric System" the authors propose a physical apparatus for the realization of a faster than Hermitian evolution. This last appears in contrast with the conclusions obtained in our paper "Some Remarks on Quantum Brachistochrone". We will clarify this apparent contradiction and some problematic aspects of the treatment in [1].
Masillo [1] commented on our manuscript [2] "Observation of a Fast Evolution in a Parity-time-symmetric System", pointing out a contradiction of our work with Ref.[3]. In this reply, we pointed out there is no disagreement between Masillo's comment and our work in Ref. [2]. The efficiency cost pointed out in Ref.\cite{masillo} exists, namely to obtain the PT-symmetric hamiltonian evolution, one has to make a measurement on the auxiliary qubit and the auxiliary qubit is at state $|0\ket$ only probabilistically. This is reflected in the amplitude of the spectrum in the NMR quantum simulation. As…
Jia and Dutra (J. Phys. A: Math. Gen. 39 (2006) 11877) have considered the one-dimensional non-Hermitian complexified potentials with real spectra in the context of position-dependent mass in Dirac equation. In their second example, a smooth step shape mass distribution is considered and a non-Hermitian non - PT- symmetric Lorentz vector potential is obtained. They have mapped this problem into an exactly solvable Rosen-Morse Schrodinger model and claimed that the energy spectrum is real. The energy spectrum they have reported is pure imaginary or at best forms an empty set. Their claim on the…
In "Comment on Supersymmetry, PT-symmetry and spectral bifurcation" \cite{BQ1}, Bagchi and Quesne correctly show the presence of a class of states for the complex Scarf-II potential in the unbroken PT-symmetry regime, which were absent in \cite{AP}. However, in the spontaneously broken PT-symmetry case, their argument is incorrect since it fails to implement the condition for the potential to be PT-symmetric: $C^{PT}[2(A-B)+α]=0$. It needs to be emphasized that in the models considered in \cite{AP}, PT is spontaneously broken, implying that the potential is PT- symmetric, whereas the ground st…
Using the Nikiforov Uvarov method, we obtained the eigenvalues and eigenfunctions of the Woods Saxon potential with the negative energy levels based on the mathematical approach. According to the PT Symmetric quantum mechanics, we exactly solved the time independent Shcrodinger equation for the same potential. Results are obtained for the s states.
Tensor eigenvalues and eigenvectors have been introduced in the recent mathematical literature as a generalization of the usual matrix eigenvalues and eigenvectors. We apply this formalism to a tensor that describes a multipartite symmetric state or a spin state, and we investigate to what extent the corresponding tensor eigenvalues contain information about the multipartite entanglement (or, equivalently, the classicality) of the state. This extends previous results connecting entanglement to spectral properties related to the state. While for spin-1 states the positivity of the smallest tens…
An analytical approximation for the eigenvalues of $\mathcal{PT}$ symmetric Hamiltonian $\mathsf{H} = -d^{2}/dx^{2} - (\mathrm{i}x)^{ε+2}$, $ε> -1$ is developed via simple basis sets of harmonic-oscillator wave functions with variable frequencies and equilibrium positions. We demonstrate that our approximation provides high accuracy for any given energy level for all values of $ε> -1$.
The spectrum of complex PT-symmetric potential, $V(x)=igx$, is known to be null. We enclose this potential in a hard-box: $V(|x| \ge 1) =\infty $ and in a soft-box: $V(|x|\ge 1)=0$. In the former case, we find real discrete spectrum and the exceptional points of the potential. The asymptotic eigenvalues behave as $E_n \sim n^2.$ The solvable purely imaginary PT-symmetric potentials vanishing asymptotically known so far do not have real discrete spectrum. Our solvable soft-box potential possesses two real negative discrete eigenvalues if $|g|<(1.22330447)$. The soft-box potential turns out to b…
We analyze the conclusions of the influence of a Coulomb-type potential on the Klein-Gordon oscillator. We show that the truncation method proposed by the authors do not yield all the eigenvalues of the radial equation but just one of them for a particular value of a model parameter. Besides, the existence of allowed oscillator frequencies that depend on the quantum numbers is an artifact of the truncation method.
In a recent Comment [arXiv: 1101.3980] Shalaby criticised our paper "Families of Particles with Different Masses in PT-Symmetric Quantum Field Theory" [arXiv:1002.3253]. On examining his arguments, we find that there are serious flaws at almost every stage of his Comment. In view of space and time considerations, we point out the major flaws that render his arguments invalid. Essentially Shalaby is attempting to obtain our results from a variational principle and to find a physical interpretation of his calculation. The variational procedure that he uses is inapplicable, and his description of…
Using Hill's determinant method we show that the set of Judd's solutions is only a subset of all the eigenvalues with the form $E_n=nω-g^2/ω$ in the spectrum of the Rabi model. Therefore Braak's solution of the quantum Rabi model is not complete.
This is a response to a recently reported comment [1] on paper [J. Math. Phys.59, 082105 (2018)] regarding the quantization of damped harmonic oscillator using a non-Hermitian Hamiltonian with real energy eigenvalues. We assert here that the calculation of Eq. (29) of [2] is incorrect, and thus the subsequent steps via the Nikiforov-Uvarov method are affected, and the energy eigenvalues should have been complex. However, we show here that the Hermiticity of the Hamiltonian should be firstly achieved to make the correct transition from classical Hamiltonian to quantum counterpart, and this can …
Comment on "Scattering of light by a parity-time-symmetric dipole beyond the first Born approximation"
本文针对一篇计算三维PT对称双δ函数势散射振幅的论文，指出其分析存在基本错误、结果无定论，并给出了该势散射振幅的精确闭式表达式。
We explain why the main conclusion of Bender et al, hep-th/0511229 [J. Phys. A 39 (2006) 1657] regarding the practical superiority of the non-Hermitian description of PT-symmetric quantum systems over their Hermitian description is not valid. Recalling the essential role played by the Hermitian description in the characterization and interpretation of the physical observables, we maintain that as far as the physical aspects of the theory are concerned the Hermitian description is not only unavoidable but also indispensable.
We prove the reality of the perturbed eigenvalues of some PT symmetric Hamiltonians of physical interest by means of stability methods. In particular we study 2-dimensional generalized harmonic oscillators with polynomial perturbation and the one-dimensional $x^2(ix)^ε$ for $-1<ε<0$.
The purpose of this paper is to study the problem of computing unitary eigenvalues (U-eigenvalues) of non-symmetric complex tensors. By means of symmetric embedding of complex tensors, the relationship between U-eigenpairs of a non-symmetric complex tensor and unitary symmetric eigenpairs (US-eigenpairs) of its symmetric embedding tensor is established. An algorithm (Algorithm \ref{algo:1}) is given to compute the U-eigenvalues of non-symmetric complex tensors by means of symmetric embedding. Another algorithm, Algorithm \ref{algo:2}, is proposed to directly compute the U-eigenvalues of non-sy…
Recently a method for adiabatic quantum computation has been proposed and there has been considerable speculation about its efficiency for NP-complete problems. Heuristic arguments in its favor are based on the unproven assumption of an eigenvalue gap. We show that, even without the assumption of an eigenvalue gap, other standard arguments can be used to show that a large class of Hamiltonians proposed for adiabatic quantum computation have unique ground states. We also discuss some of the issues which arise in trying to analyze the behavior of the eigenvalue gap. In particular, we propose sev…
Reply to the Comment on `The operational foundations of PT-symmetric and quasi-Hermitian quantum theory'
本文回复Znojil对PT对称与准厄米量子理论操作基础的评论，澄清动机问题遭误解，说明文献取舍理由，并讨论标准量子理论扩展的可行约束与前景。
The spectra of a particular class of PT symmetric eigenvalue problems has previously been studied, and found to have an extremely rich structure. In this paper we present an explanation for these spectral properties in terms of quantisation conditions obtained from the complex WKB method. In particular, we consider the relation of the quantisation conditions to the reality and positivity properties of the eigenvalues. The methods are also used to examine further the pattern of eigenvalue degeneracies observed by Dorey et al. in [1,2].
In this comment, we show that the solutions of the-wave Dirac equation for deformed generalized Pöschl-Teller potential obtained by Wei et al are valid only for $q\geq1$ and $\frac{1}{2α}\ln{q}<r<\infty$. When $0<q<1$, we prove that the energy eigenvalues for the bound states are given by the solution of a transcendental equation involving the hypergeometric function. To test our results, the Morse potential is considered as limiting case.
Eigenvalue attraction in open quantum systems, biophysical systems, and Parity-Time symmetric materials
本文研究开放量子系统、生物物理系统及PT对称材料中的本征值吸引现象，通过推导实矩阵本征值二阶导数表达式，揭示了惯性力、复共轭本征值间吸引及谱中其他本征值作用力的贡献。
The paper by Bowen, Mancini, Fessatidis, and Murawski (2012 Phys. Scr. {\bf 85}, 065005) demonstrates in a dramatic fashion the serious difficulties that can arise when one rushes to perform numerical studies before understanding the physics and mathematics of the problem at hand and without understanding the limitations of the numerical methods used. Based on their flawed numerical work, the authors conclude that the work of Bender and Boettcher is wrong even though it has been verified at a completely rigorous level. Unfortunately, the numerical procedures performed and described in the pape…
A recent paper [J. Math. Phys. {\bf 59}, 082105 (2018)] constructs a Hamiltonian for the (dissipative) damped harmonic oscillator. We point out that non-Hermiticity of this Hamiltonian has been ignored to find real discrete eigenvalues which are actually non-real. We emphasize that non-Hermiticity in Hamiltonian is crucial and it is a quantal signature of dissipation.
In this paper, a non-trivial system governed by a continuum PT-symmetric Hamiltonian is discussed. We show that this Hamiltonian is iso-spectral to the simple harmonic oscillator. We find its eigenfunctions and the path in the complex plane along which these functions form an orthonormal set. We also find the hidden symmetry operator, ${\cal C}$, for this system. All calculations are performed analytically and without approximation.
An elucidation of the current state of art in quasi-Hermitian quantum theory (QHQT) as inspired by the recent paper by Alase et al (J. Phys. A: Math. Theor. 55 (2022) 244003, paper [1]) is offered. We point out that the author's main discovery (viz., that the QHQT does not extend the standard quantum theory) is not new. In a related comment on the author's method of proof performed in ``the framework of general probabilistic theories'' (GPT) we add that also in this context a few other, mathematically consistent GPT-like theories are already available in the literature (pars pro toto we mentio…
We consider a class of one-dimensional nonhermitian oscillators and discuss the relationship between the real eigenvalues of PT-symmetric oscillators and the resonances obtained by different authors. We also show the relationship between the strong-coupling expansions for the eigenvalues of those oscillators. Comparison of the results of the complex rotation and the Riccati-Padé methods reveals that the optimal rotation angle converts the oscillator into either a PT-symmetric or an Hermitian one. In addition to the real positive eigenvalues the PT-symmetric oscillators exhibit real positive re…
We present a large class of non-Hermitian non-PT-symmetric two-component Dirac Hamiltoninas with real energy spectra. These Hamiltonians are invariant under the combined action of "charge" conjugation (two-component transpose) and space-parity. Examples are given from the two subclasses of these systems having localized and/or continuum states with real energies.
Many eigenvalue problems arising in practice are often of the generalized form $A\x=λB\x$. One particularly important case is symmetric, namely $A, B$ are Hermitian and $B$ is positive definite. The standard algorithm for solving this class of eigenvalue problems is to reduce them to Hermitian eigenvalue problems. For a quantum computer, quantum phase estimation is a useful technique to solve Hermitian eigenvalue problems. In this work, we propose a new quantum algorithm for symmetric generalized eigenvalue problems using ordinary differential equations. The algorithm has lower complexity than…
Extending the supersymmetric method proposed by Tkachuk to the complex domain, we obtain general expressions for superpotentials allowing generation of quasi-exactly solvable PT-symmetric potentials with two known real eigenvalues (the ground state and first-excited state energies). We construct examples, namely those of complexified non-polynomial oscillators and of a complexified hyperbolic potential, to demonstrate how our scheme works in practice. For the former we provide a connection with the sl(2) method, illustrating the comparative advantages of the supersymmetric one.
The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian $H=p^2+x^2(ix)^ε$ has real, positive, and discrete eigenvalues for all $ε\geq 0$. These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues $E_n=2n+1$ (n=0, 1, 2, 3, ...) at $ε=0$. However, the harmonic oscillator also has negative eigenvalues $E_n=-2n-1$ (n=0, 1, 2, 3, ...), and one may ask whether it is equally possible to continue analytically from these eigenvalues. It is shown in this paper that for appropriate PT-symmetric boundary conditions the Hamiltonian $H=p^2+x^2(ix)^ε$ also has real and {\it ne…
We analyze a set of three PT-symmetric complex potentials, namely harmonic oscillator, generalized Poschl-Teller and Scarf II, all of which reveal a double series of energy levels along with the corresponding superpotential. Inspired by the fact that two superpotentials reside naturally in order-two parasupersymmetry (PSUSY) and second-derivative supersymmetry (SSUSY) schemes, we complexify their frameworks to successfully account for the three potentials.
We study a non-Hermitian $PT-$symmetric generalization of an $N$-particle, two-mode Bose-Hubbard system, modeling for example a Bose-Einstein condensate in a double well potential coupled to a continuum via a sink in one of the wells and a source in the other. The effect of the interplay between the particle interaction and the non-Hermiticity on characteristic features of the spectrum is analyzed drawing special attention to the occurrence and unfolding of exceptional points (EPs). We find that for vanishing particle interaction there are only two EPs of order $N+1$ which under perturbation u…
We study the eigenvalue problem $-u^{\prime\prime}(z)-[(iz)^m+P_{m-1}(iz)]u(z)=λu(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\fracπ{2}\pm \frac{2π}{m+2}$, where $P_{m-1}(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial and integers $m\geq 3$. We provide an asymptotic expansion of the eigenvalues $λ_n$ as $n\to+\infty$, and prove that for each {\it real} polynomial $P_{m-1}$, the eigenvalues are all real and positive, with only finitely many exceptions.
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