B. Bagchi, C. Quesne, R. Roychoudhury
We exploit the hidden symmetry structure of a recently proposed non-Hermitian Hamiltonian and of its Hermitian equivalent one. This sheds new light on the pseudo-Hermitian character of the former and allows access to a generalized quantum condition. Special cases lead to hyperbolic and Morse-like potentials in the framework of a coordinate-dependent mass model.
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We discuss certain features of pseudo-Hermiticity and weak pseudo-Hermiticity conditions and point out that, contrary to a recent claim, there is no inconsistency if the correct orthogonality condition is used for the class of pseudo-Hermitian, PT-symmetric Hamiltonians of the type $H_β = [p + {\rm i} βν(x)]^2/2m + V(x)$.
We study a two-dimensional exactly solvable non-Hermitian $PT-$non-symmetric quantum model with real spectrum, which is not amenable to separation of variables, by supersymmetrical methods. Here we focus attention on the property of pseudo-Hermiticity, biorthogonal expansion and pseudo-metric operator. To our knowledge this is the first time that pseudo-Hermiticity is realized explicitly for a nontrivial two-dimensional case. It is shown that the Hamiltonian of the model is not diagonalizable.
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-compon…
We give a necessary and sufficient condition for the reality of the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors.
When a non-hermitian hamiltonian has a certain symmetry, such as the PT pseudo-hermiticity, it is still possible that the hamiltonian has a real spectrum. In this note, by adding an imaginary potential proportional to ip_1p_2 to the hamiltonian of an anisotropic planar oscillator, we construct a PT pseudo-hermitian hamiltonian and obtain its real spectrum. In addition, we find that the reality of the energy spectrum remains when the model is extended to the canonical noncommutative space.
We study SUSY-intertwining for non-Hermitian Hamiltonians with special emphasis to the two-dimensional generalized Morse potential, which does not allow for separation of variables. The complexified methods of SUSY-separation of variables and two-dimensional shape invariance are used to construct particular solutions - both for complex conjugated energy pairs and for non-paired complex energies.
Parallels between the notions of nonlinear pseudobosons and of an apparent non-Hermiticity of observables as shown in paper I (arXiv: 1109.0605) are demonstrated to survive the transition to the quantum models based on the use of unbounded metric in the Hilbert space of states.
For a given standard Hamiltonian H=[p-A(x)]^2/(2m)+V(x) with arbitrary complex scalar potential V and vector potential A, with x real, we construct an invertible antilinear operator τsuch that H is τ-anti-pseudo-Hermitian, i.e., H^\dagger=τHτ^{-1}. We use this result to give the explicit form of a linear Hermitian invertible operator with respect to which any standard PT-symmetric Hamiltonian with a real degree of freedom is pseudo-Hermitian. Our results do not make use of the assumption that H is diagonalizable or that its spectrum is discrete.
We study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quant-ph/0208076) on the CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we introduce generalized parity (${\cal P}$), time-reversal (${\cal T}$), and charge-conj…
Pseudo-Hermitian operators can be used in modeling electromagnetic wave propagation in stationary lossless media. We extend this method to a class of non-dispersive anisotropic media that may display loss or gain. We explore three concrete models to demonstrate the utility of our general results and reveal the physical meaning of pseudo-Hermiticity and quasi-Hermiticity of the relevant wave operator. In particular, we consider a uniaxial model where this operator is not diagonalizable. This implies left-handedness of the medium in the sense that only clockwise circularly polarized plane-wave s…
For a weakly pseudo-Hermitian linear operator, we give a spectral condition that ensures its pseudo-Hermiticity. This condition is always satisfied whenever the operator acts in a finite-dimensional Hilbert space. Hence weak pseudo-Hermiticity and pseudo-Hermiticity are equivalent in finite-dimensions. This equivalence extends to a much larger class of operators. Quantum systems whose Hamiltonian is selected from among these operators correspond to pseudo-Hermitian quantum systems possessing certain symmetries.
For an invertible (bounded) linear operator Q acting in a Hilbert space ${\cal H}$, we consider the consequences of the QT-symmetry of a non-Hermitian Hamiltonian $H:{\cal H}\to{\cal H}$ where T is the time-reversal operator. If H is symmetric in the sense that ${\cal T} H^\dagger {\cal T}=H$, then QT-symmetry is equivalent to Q^{-1}-weak-pseudo-Hermiticity. But in general this equivalence does not hold. We show this using some specific examples. Among these is a large class of non-PT-symmetric Hamiltonians that share the spectral properties of PT-symmetric Hamiltonians.
The increasingly popular concept of a hidden Hermiticity of operators (i.e., of their Hermiticity with respect to an {\it ad hoc} inner product in Hilbert space) is compared with the recently introduced notion of {\em non-linear pseudo-bosons}. The formal equivalence between these two notions is deduced under very general assumptions. Examples of their applicability in quantum mechanics are discussed.
We show that a diagonalizable (non-Hermitian) Hamiltonian H is pseudo-Hermitian if and only if it has an antilinear symmetry, i.e., a symmetry generated by an invertible antilinear operator. This implies that the eigenvalues of H are real or come in complex conjugate pairs if and only if H possesses such a symmetry. In particular, the reality of the spectrum of H implies the presence of an antilinear symmetry. We further show that the spectrum of H is real if and only if there is a positive-definite inner-product on the Hilbert space with respect to which H is Hermitian or alternatively there …
Pseudo-Hermiticity, Anti-Pseudo-Hermiticity, and Generalized Parity-Time-Reversal Symmetry at Exceptional Points
本文将可对角化线性算子伪厄米性与广义宇称-时间反演（$PT$）对称性的等价性推广至块可对角化算子，并给出二维和无限维希尔伯特空间中伪厄米算子在例外点处及远离时实现相关对称性的反线性算子显式表达式。
We survey some of the main conceptual developments in the study of PT-symmetric and pseudo-Hermitian Hamiltonian operators that have taken place during the past ten years or so. We offer a precise mathematical description of a quantum system and its representations that allows us to describe the idea of unitarization of a quantum system by modifying the inner product of the Hilbert space. We discuss the role and importance of the quantum-to-classical correspondence principle that provides the physical interpretation of the observables in quantum mechanics. Finally, we address the problem of co…
We consider the special type of pseudo-bosonic systems that can be mapped to standard bosons by means of generalized Bogoliubov transformation and demonstrate that a pseudo-Hermitian systems can be obtained from them by means of a second subsequent Bogoliubov transformation. We employ these operators in a simple model and study three different types of scenarios for the constraints on the model parameters giving rise to a Hermitian system, a pseudo-Hermitian system in which the second the Bogoliubov transformations is equivalent to the associated Dyson map and one in which we obtain D-quasi ba…
A class of spherically symmetric non-Hermitian Hamiltonians and their η-weak-pseudo-Hermiticity generators are presented. An operators-based procedure is introduced so that the results for the 1D Schrodinger Hamiltonian may very well be reproduced. A generalization beyond the nodeless states is proposed. Our illustrative examples include η-weak-pseudo-Hermiticity generators for the non-Hermitian weakly perturbed 1D and radial oscillators, the non-Hermitian perturbed radial Coulomb, and the non-Hermitian radial Morse models.
We inquire into some properties of diagonalizable pseudo-Hermitian operators, showing that their definition can be relaxed and that the pseudo-Hermiticity property is strictly connected with the existence of an antilinear symmetry. This result is then illustrated by considering the particular case of the complex Morse potential.
Exact solvability of some non-Hermitian $η$-weak-pseudo-Hermitian Hamiltonians is explored as a byproduct of $η$-weak-pseudo-Hermiticity generators. A class of V_{eff}(x)=V(x)+iW(x) potentials is considered, where the imaginary part W(x) is used as an $η$-weak-pseudo-Hermiticity generator to obtain exactly solvable $η$-weak-pseudo-Hermitian Hamiltonian models.
We examine the possibility of artificial Hawking radiation by proposing a non-PT-symmetric weakly pseudo-Hermitian two-band model containing a tilting parameter by pursuing Weyl semimetal blackhole analogy. We determine the tunnelling probability using such a Hamiltonian through the event horizon that acts as a classically forbidden barrier.
We give two characterization theorems for pseudo-Hermitian (possibly nondiagonalizable) Hamiltonians with a discrete spectrum that admit a block-diagonalization with finite-dimensional diagonal blocks. In particular, we prove that for such an operator H the following statements are equivalent. 1. H is pseudo-Hermitian; 2. The spectrum of H consists of real and/or complex-conjugate pairs of eigenvalues and the geometric multiplicity and the dimension of the diagonal blocks for the complex-conjugate eigenvalues are identical; 3. H is Hermitian with respect to a positive-semidefinite inner produc…
The Hermitian systems possess unitary scattering; however, the Hermiticity is unnecessary for a unitary scattering although the scattering under the influence of non-Hermiticity is mostly non-unitary. Here we prove that the unitary scattering is protected by certain type of pseudo-Hermiticity and unaffected by the degree of non-Hermiticity. The energy conservation is violated in the scattering process and recovers after scattering. The subsystem of the pseudo-Hermitian scattering center including only the connection sites is Hermitian. These findings provide fundamental insights on the unitary…
It is generally believed that Parity-Time (PT)-symmetry breaking occurs when eigenvalues or both eigenvalues and eigenvectors coincide. However, we show that this well-accepted picture of PT-symmetry breaking is incorrect. Instead, we demonstrate that the physical mechanism of PT-symmetry breaking is the resonance between positive- and negative-action modes. It is proved that PT-symmetry breaking occurs when and only when this resonance condition is satisfied, and this mechanism applies to all known PT-symmetry breakings observed in different branches of physics. The result is achieved by prov…
We prove that in finite dimensions, a Parity-Time (PT)-symmetric Hamiltonian is necessarily pseudo-Hermitian regardless of whether it is diagonalizable or not. This result is different from Mostafazadeh's, which requires the Hamiltonian to be diagonalizable. PT-symmetry breaking often occurs at exceptional points where the Hamiltonian is not diagonalizable. Our result implies that PT-symmetry breaking is equivalent to the onset of instabilities of pseudo-Hermitian systems, which was systematically studied by Krein et al. in 1950s. In particular, we show that the mechanism of PT-symmetry breaki…
A class of non-Hermitian d-dimensional Hamiltonias with position dependent mass and their $η$-pseudo-Hermiticity generators is presented. Illustrative examples are given in 1D, 2D, and 3D for different position dependent mass settings.
The recently introduced two- and three-parameter ($p,q$)- and ($p,q,μ$)-deformed extensions of the Heisenberg algebra were explored under the condition of their connectedness with the respective nonstandard (other than known ones) deformed quantum oscillator algebras. In this paper we show that such connection dictates certain new $η(N)$-pseudo-Hermitian conjugation rule between the creation and annihilation operators, with $η(N)$ depending on the particle number operator $N$. In turn, that leads to the related $η(N)$-pseudo-Hermiticity of the position/momentum operators, though the involved H…
The spectral branching behavior of the 2x2 operator matrix of the magneto-hydrodynamic alpha^2-dynamo is analyzed numerically. Some qualitative aspects of level crossings are briefly discussed with the help of a simple toy model which is based on a Z_2-graded-pseudo-Hermitian 2x2 matrix. The considered issues comprise: the underlying SU(1,1) symmetry and the Krein space structure of the system, exceptional points of branching type and diabolic points, as well as the algebraic and geometric multiplicity of corresponding degenerate eigenvalues.
We propose that the real spectrum and the orthogonality of the states for several known complex potentials of both types, PT-symmetric and non-PT-symmetric can be understood in terms of currently proposed $η$-pseudo-Hermiticity (Mostafazadeh, quant-ph/0107001) of a Hamiltonian, provided the Hermitian linear automorphism, $η$, is introduced as $e^{-θp}$ which affects an imaginary shift of the co-ordinate : $e^{-θp} x e^{θp} = x+iθ$.
It is known that the standard and the inverted harmonic oscillator are different. Replacing thus of ω by iω in the regular oscillator is necessary going to give the inverted oscillator H^{r}. This replacement would lead to anti- PT-symmetric harmonic oscillator Hamiltonian (iH^{os}). The pseudo-hermiticity relation has been used here to relate the anti-PT-symmetric harmonic Hamiltonian to the inverted oscillator. By using a simple algebra, we introduce the ladder operators describing the inverted harmonic oscillator to reproduce the analytical solutions.We construct the inverted coherent state…
The main achievements of Pseudo-Hermitian Quantum Mechanics and its distinction with the indefinite-metric quantum theories are reviewed. The issue of the non-uniqueness of the metric operator and its consequences for defining the observables are discussed. A systematic perturbative expression for the most general metric operator is offered and its application for a toy model is outlined.
Pseudo-Hermitian operators appear in the solution of Maxwell's equations for stationary non-dispersive media with arbitrary (space-dependent) permittivity and permeability tensors. We offer an extension of the results in this direction to certain stationary dispersive media. In particular, we use the WKB approximation to derive an explicit expression for the planar time-harmonic solutions of Maxwell's equations in an inhomogeneous dispersive medium and study the combined affect of inhomogeneity and dispersion.
Recently some authors have broadened the scope of canonical quantum mechanics by replacing the conventional Hermiticity condition on the Hamiltonian by a weaker requirement through the introduction of the notion of pseudo-Hermiticity. In the present study we investigate eigenvalues, transmission and reflection from complex optical potentials enjoying the property of pseudo-Hermiticity.
Constructible reality condition of pseudo entropy via pseudo-Hermiticity
本文应用pseudo-Hermiticity概念构建了pseudo entropy的实数条件，找到了转移矩阵的一般形式，并证明其与量子场论的Tomita-Takesaki模理论相关。
Emphasizing the physical constraints on the formulation of a quantum theory based on the standard measurement axiom and the Schroedinger equation, we comment on some conceptual issues arising in the formulation of PT-symmetric quantum mechanics. In particular, we elaborate on the requirements of the boundedness of the metric operator and the diagonalizability of the Hamiltonian. We also provide an accessible account of a Krein-space derivation of the CPT-inner product that was widely known to mathematicians since 1950's. We show how this derivation is linked with the pseudo-Hermitian formulati…
Motivated by the recent developments of pseudo-hermitian quantum mechanics, we analyze the structure of unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (PIP space). Next, we introduce several generalizations of the notion of similarity between operators and explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular PIP space, namely, a scale of Hilbert spaces generated by a metric opera…
We express the electromagnetic field propagating in an arbitrary time-independent non-dispersive medium in terms of an operator that turns out to be pseudo-Hermitian for Hermitian dielectric and magnetic permeability tensors Epsilon and Mu. We exploit this property to determine the propagating field. In particular, we obtain an explicit expression for a planar field in an isotropic medium with Epsilon and Mu varying along the direction of the propagation. We also study the scattering of plane waves due to a localized inhomogeneity.
Pseudo-Hermiticity protects the energy-difference conservation in the scattering
本研究发现非厄米散射系统中由赝厄米性保护的能量差守恒律，通过两个典型反PT对称系统验证了该守恒律的成立与失效，为非厄米系统的相关物理研究提供了新见解。
We employ a recently-developed transfer-matrix formulation of scattering theory in two dimensions to study a class of scattering setups modeled by real potentials. The transfer matrix for these potentials is related to the time-evolution operator for an associated pseudo-Hermitian Hamiltonian operator $\widehat{\mathbf{H}}$ which develops an exceptional point for a discrete set of incident wavenumbers. We use the spectral properties of this operator to determine the transfer matrix of these potentials and solve their scattering problem. We apply our general results to explore the scattering of…
A complexified von Roos Hamiltonian is considered and a Hermitian first-order intertwining differential operator is used to obtain the related position dependent mass $η$-weak-pseudo-Hermitian Hamiltonians. Using a Liouvillean-type change of variables, the $η$-weak-pseudo-Hermitian von Roos Hamiltonians H(x) are mapped into the traditional Schrodinger Hamiltonian form H(q), where exact isospectral correspondence between H(x) and H(q) is obtained. Under a user-friendly position dependent mass settings, it is observed that for each exactly-solvable $η$-weak-pseudo-Hermitian reference-Hamiltonian…
Pseudo entropy and pseudo-Hermiticity in quantum field theories
本文在量子场论中研究伪Rényi熵，分析了不同算符分布下其对数项的实复性与伪厄米条件的关联，并在二维共形场论中发现二阶伪Rényi熵存在仅依赖算符共形维数的普适发散项，高阶发散项则与理论细节相关。
For a given pseudo-Hermitian Hamiltonian of the standard form: H=p^2/2m+v(x), we reduce the problem of finding the most general (pseudo-)metric operator ηsatisfying H^\dagger=ηH η^{-1} to the solution of a differential equation. If the configuration space is the real line, this is a Klein-Gordon equation with a nonconstant mass term. We obtain a general series solution of this equation that involves a pair of arbitrary functions. These characterize the arbitrariness in the choice of η. We apply our general results to calculate ηfor the PT-symmetric square well, an imaginary scattering potentia…
A Hermitian and an anti-Hermitian first-order intertwining operators are introduced and a class of $η$-weak-pseudo-Hermitian position-dependent mass (PDM) Hamiltonians are constructed. A corresponding reference-target $η$-weak-pseudo-Hermitian PDM -- Hamiltonians' map is suggested. Some $η$-weak-pseudo-Hermitian PT -symmetric Scarf II and periodic-type models are used as illustrative examples. Energy-levels crossing and flown-away states phenomena are reported for the resulting Scarf II spectrum. Some of the corresponding $η$-weak-pseudo-Hermitian Scarf II- and periodic-type-isospectral models…
Deformed Heisenberg algebra and its Hilbert space representations
本文提出一种变形的海森堡代数，通过伪相似性变换证明其厄米克性，并研究了变形空间中自由粒子的本征值问题。
We introduce generalized versions of complex Scarf and Morse-type potentials that con- tain energy-dependent parameters. PT -symmetry and pseudo-hermiticity of the associated quantum systems are discussed, and a modified orthogonality relation and pseudo-norm are constructed. We show that despite energy-dependence, our systems can admit real energy spectra and normalizable solutions of bound-state type.
A general strategy is provided to identify the most general metric for diagonalizable pseudo-Hermitian and anti-pseudo-Hermitian Hamilton operators represented by two-dimensional matrices. It is investigated how a permutation of the eigen-values of the Hamilton operator in the process of its diagonalization influences the metric and how this permutation equivalence affects energy eigen-values. We try to understand on one hand, how the metric depends on the normalization of the chosen left and right eigen-basis of the matrix representing the diagonalizable pseudo-Hermitian or anti-pseudo-Hermit…
Non-Hermiticity induced thermal entanglement phase transition
研究通过非厄运性在无外磁场下诱导两量子比特系统的最大纠缠和量子相变，揭示非厄运性驱动的能隙关闭导致纠缠相变，不同于例外点。
We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT-symmetric quantum mechanics.
On the η pseudo PT symmetry theory for non-Hermitian Hamiltonians: time-dependent systems
本文针对含时非厄米哈密顿量，提出一种新度量η(t)=PTη(t)以同时推导伪PT对称性与η伪厄米性，求解SU(1,1)模型并给出物理应用。
Pseudo-hermitian Chebyshev differential matrix and non-Hermitian Liouville quantum mechanics
本研究聚焦谱配置法中切比雪夫微分矩阵的伪厄米性引发的本征态完备性丧失等问题，重新审视非厄米刘维尔量子力学并探索其扩展模型，同时证实紧化参数会影响谱不稳定性。
Non-Hermiticity can vary the topology of system, induce topological phase transition, and even invalidate the conventional bulk-boundary correspondence. Here, we show the introducing of non-Hermiticity without affecting the topological properties of the original chiral symmetric Hermitian systems. Conventional bulk-boundary correspondence holds, topological phase transition and the (non)existence of edge states are unchanged even though the energy bands are inseparable due to non-Hermitian phase transition. Chern number for energy bands of the generalized non-Hermitian system in two dimension …
Non-Hermiticity enhanced topological immunity of one-dimensional $p$-wave superconducting chain
本文研究一维广义 Kitaev 模型，证明非对称跃迁引入的非厄米性可扩大 Majorana 零模抵御非局域无序的参数区，并稳定拓扑超导相。
We show that complex Lie algebras (in particular sl(2,C)) provide us with an elegant method for studying the transition from real to complex eigenvalues of a class of non-Hermitian Hamiltonians: complexified Scarf II, generalized Pöschl-Teller, and Morse. The characterizations of these Hamiltonians under the so-called pseudo-Hermiticity are also discussed.
In this paper we present a general method to solve non hermetic potentials with PT symmetry using the introduction of two first-order operator against η-pseudo-hermetic(η-weak-pseudo-hermiticity) with position dependent effective mass and applying it to Dirac equation with spinor wave function and non-hermitic potentials with real energy spectrum considering position dependent effective mass. Also Pöschl-teller complex and Eckart potentials will be discussed as examples.
Pseudo-real quantum fields
本文引入复数的伪实性概念，为伪厄米量子场论的两种方法提供统一框架，解决了规范理论表述问题，并推动了其在弯曲时空中的发展。
Locality and Exceptional Points in Pseudo-Hermitian Physics
本论文研究拟厄米与赝厄米物理，证明局部可观测量及非定域关联边界受限，并分析例外点、非厄米缺陷晶格、PT 相变及互缠算子构造。
We consider a Bogolibov-de Geenes (BdG) Hamiltonian, which is a non-Hermitian Hamiltonian with pseudo-Hermiticity, for a system of (pseudo) spin-$1/2$ bosons in a honeycomb lattice under the condition that the population difference between the two spin components, i.e., magnetization, is a constant. Such a system is capable of acting as a topological amplifier, under time-reversal symmetry, with stable bulk bands but unstable edge modes which can be populated at an exponentially fast rate. We quantitatively study the topological properties of this model within the framework of the 38-fold way …
Degeneracies and symmetry breaking in pseudo-Hermitian matrices
本文通过分类赝厄米矩阵的本征空间，证明对称性破缺当且仅当相反类型本征值在实轴碰撞时发生，并据此刻画了参数空间中稳定相区域、例外曲面及赝厄米性破缺点的完整结构。
According to von Neumann, the global Hamiltonian of whole universe must be Hermitian in order to keep the eigenvalues real and to construct a self-consistent quantum theory. In addition to the open system approach by introducing environmental degrees of freedom to a small system, a global Hermitian Hamiltonian can also be generated through the dilation from a small Hilbert space. For example, a local non-Hermitian $\cal PT$-symmetric system can be simulated with a global Hermitian one by the Naimark dilation. When shared by Alice and Bob, the internal nonlocality in such dilated Hermitian syst…
We investigate the effects of non-Hermiticity on interacting fermionic systems. We do this by including non-Hermitian bilinear terms into the 3+1 dimensional Nambu--Jona-Lasinio (NJL) model. Two possible bilinear modifications give rise to $\mathcal{PT}$ symmetric theories; this happens when the standard NJL model is extended either by a pseudovector background field $ig \barψγ_5 B_μγ^μψ$ or by an antisymmetric-tensor background field $g \barψF_{μν}γ^μγ^νψ$. The three remaining bilinears are {\it anti}-$\mathcal{PT}$-symmetric in nature, $ig \barψB_μγ^μψ, ig\barψγ_5 ψ$ and $ig\barψ{1}ψ$, so th…
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