Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries

We extend the definition of generalized parity $P$, charge-conjugation $C$ and time-reversal $T$ operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and we use these generalized operators to describe the full set of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold classification. In particular we show that $TP$ and $CTP$ are the generators of the antiunitary symmetries; moreover, a necessary and sufficient condition is provided for a pseudo-Hermitian Hamiltonian $H$ to admit a $P$-reflecting symmetry which generates the $P$-pseudounitary and the $P$-pseudoantiunitary symmetries. Finally, a physical example is considered and some hints on the $P$-unitary evolution of a physical system are also given.Comment: 20 page

Generalized Swanson models and their solutions

We analyze a class of non-Hermitian quadratic Hamiltonians, which are of the form $H = {\cal{A}}^{\dagger} {\cal{A}} + \alpha {\cal{A}} ^2 + \beta {\cal{A}}^{\dagger 2}$, where $\alpha, \beta$ are real constants, with $\alpha \neq \beta$, and ${\cal{A}}^{\dagger}$ and ${\cal{A}}$ are generalized creation and annihilation operators. Thus these Hamiltonians may be classified as generalized Swanson models. It is shown that the eigenenergies are real for a certain range of values of the parameters. A similarity transformation $\rho$, mapping the non-Hermitian Hamiltonian $H$ to a Hermitian one $h$, is also obtained. It is shown that $H$ and $h$ share identical energies. As explicit examples, the solutions of a couple of models based on the trigonometric Rosen-Morse I and the hyperbolic Rosen-Morse II type potentials are obtained. We also study the case when the non-Hermitian Hamiltonian is ${\cal{PT}}$ symmetric.Comment: 17 page

Pseudo-Unitary Operators and Pseudo-Unitary Quantum Dynamics

We consider pseudo-unitary quantum systems and discuss various properties of pseudo-unitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudo-unitary operators with finite-dimensional diagonal blocks. Furthermore, we show that every pseudo-unitary matrix is the exponential of $i=\sqrt{-1}$ times a pseudo-Hermitian matrix, and determine the structure of the Lie groups consisting of pseudo-unitary matrices. In particular, we present a thorough treatment of $2\times 2$ pseudo-unitary matrices and discuss an example of a quantum system with a $2\times 2$ pseudo-unitary dynamical group. As other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics, demonstrate the coincidence of the symplectic group $Sp(2n)$ with the real subgroup of a matrix group that is isomorphic to the pseudo-unitary group U(n,n), and elaborate on an approach to second quantization that makes use of the underlying pseudo-unitary dynamical groups.Comment: Revised and expanded version, includes an application to symplectic transformations and groups, accepted for publication in J. Math. Phy

New features of scattering from a one-dimensional non-Hermitian (complex) potential

For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time-reversal: $R(-k)\ne R(k)$ and $T(-k) \ne T(k)$, unless the potentials are real or PT-symmetric. For complex PT-symmetric scattering potentials, we propose that $R_{left}(-k)=R_{right}(k)$ and $T(-k)=T(k)$. So far, the spectral singularities (SS) of a one-dimensional non-Hermitian scattering potential are witnessed/conjectured to be at most one. We present a new non-Hermitian parametrization of Scarf II potential to reveal its four new features. Firstly, it displays the just acclaimed (in)variances. Secondly, it can support two spectral singularities at two pre-assigned real energies ($E_*=\alpha^2,\beta^2$) either in $T(k)$ or in $T(-k)$, when $\alpha\beta>0$. Thirdly, when $\alpha \beta <0$ it possesses one SS in $T(k)$ and the other in $T(-k)$. Fourthly, when the potential becomes PT-symmetric $[(\alpha+\beta)=0]$, we get $T(k)=T(-k)$, it possesses a unique SS at $E=\alpha^2$ in both $T(-k)$ and $T(k)$. Lastly, for completeness, when $\alpha=i\gamma$ and $\beta=i\delta$, there are no SS, instead we get two negative energies $-\gamma^2$ and $-\delta^2$ of the complex PT-symmetric Scarf II belonging to the two well-known branches of discrete bound state eigenvalues and no spectral singularity exists in this case. We find them as $E^{+}_{M}=-(\gamma-M)^2$ and $E^{-}_{N}=-(\delta-N)^2$; $M(N)=0,1,2,...$ with $0 \le M (N)< \gamma (\delta)$. {PACS: 03.65.Nk,11.30.Er,42.25.Bs}Comment: 10 pages, one Table, one Figure, important changes, appeared as an FTC (J. Phys. A: Math. Theor. 45(2012) 032004

Diverse Accounting Standards on Disclosures of Islamic Financial Transactions: Prospects and Challenges of Narrowing Gaps

Purpose: Since International Financial Reporting Standards (IFRS) are not primarily meant for the accounting needs of Islamic banks, the Accounting and Auditing Organisation for Islamic Financial Institutions (AAOIFI) was established to develop specific accounting standards for Shari’ah compliance. The purpose of this paper is to assess the de jure harmonisation between the disclosure requirements of the IFRS-based Malaysian Accounting Standards (MAS) and those of the AAOIFI. Design/methodology/approach: Using Malaysia as a case study, the paper examines the extent of the de jure congruence between the IFRS-based MAS and AAOIFI’s Financial Accounting Standard No 1 (FAS1), which is considered to be one of the key disclosure standards for Islamic banks. We employ leximetrics and content analysis to analyse these accounting standards and the additional guidelines introduced by the Malaysian Accounting Standards Board (MASB) and the Central Bank of Malaysia (Bank Negara Malaysia, BNM) to identify the gaps between different tiers of MAS and FAS1. Findings: The study finds that de jure congruence between the IFRS-based MAS and AAOIFI standards has improved through the introduction of additional accounting guidelines by both the MASB and the banking regulator, BNM. However, some gaps remain between the two standards. These gaps may be difficult to completely eliminate due to differences in the fundamental principles underlying the development of both standards. Originality/value: While some studies have explored the de facto congruence between AAOIFI accounting standards and others, this paper is the first, to the best of the authors’ knowledge, to examine the de jure congruence between those standards with the IFRS-based MAS

Entangled Quantum State Discrimination using Pseudo-Hermitian System

We demonstrate how to discriminate two non-orthogonal, entangled quantum state which are slightly different from each other by using pseudo-Hermitian system. The positive definite metric operator which makes the pseudo-Hermitian systems fully consistent quantum theory is used for such a state discrimination. We further show that non-orthogonal states can evolve through a suitably constructed pseudo-Hermitian Hamiltonian to orthogonal states. Such evolution ceases at exceptional points of the pseudo-Hermitian system.Comment: Latex, 9 pages, 1 figur

Caspase-2 is upregulated after sciatic nerve transection and its inhibition protects dorsal root ganglion neurons from Apoptosis after serum withdrawal

Sciatic nerve (SN) transection-induced apoptosis of dorsal root ganglion neurons (DRGN) is one factor determining the efficacy of peripheral axonal regeneration and the return of sensation. Here, we tested the hypothesis that caspase-2(CASP2) orchestrates apoptosis of axotomised DRGN both in vivo and in vitro by disrupting the local neurotrophic supply to DRGN. We observed significantly elevated levels of cleaved CASP2 (C-CASP2), compared to cleaved caspase-3 (C-CASP3), within TUNEL+DRGN and DRG glia (satellite and Schwann cells) after SN transection. A serum withdrawal cell culture model, which induced 40% apoptotic death in DRGN and 60% in glia, was used to model DRGN loss after neurotrophic factor withdrawal. Elevated C-CASP2 and TUNEL were observed in both DRGN and DRG glia, with C-CASP2 localisation shifting from the cytosol to the nucleus, a required step for induction of direct CASP2-mediated apoptosis. Furthermore, siRNAmediated downregulation of CASP2 protected 50% of DRGN from apoptosis after serum withdrawal, while downregulation of CASP3 had no effect on DRGN or DRG glia survival. We conclude that CASP2 orchestrates the death of SN-axotomised DRGN directly and also indirectly through loss of DRG glia and their local neurotrophic factor support. Accordingly, inhibiting CASP2 expression is a potential therapy for improving both the SN regeneration response and peripheral sensory recovery

Delta-Function Potential with a Complex Coupling

We explore the Hamiltonian operator H=-d^2/dx^2 + z \delta(x) where x is real, \delta(x) is the Dirac delta function, and z is an arbitrary complex coupling constant. For a purely imaginary z, H has a (real) spectral singularity at E=-z^2/4. For \Re(z)<0, H has an eigenvalue at E=-z^2/4. For the case that \Re(z)>0, H has a real, positive, continuous spectrum that is free from spectral singularities. For this latter case, we construct an associated biorthonormal system and use it to perform a perturbative calculation of a positive-definite inner product that renders H self-adjoint. This allows us to address the intriguing question of the nonlocal aspects of the equivalent Hermitian Hamiltonian for the system. In particular, we compute the energy expectation values for various Gaussian wave packets to show that the non-Hermiticity effect diminishes rapidly outside an effective interaction region.Comment: Published version, 14 pages, 2 figure

η\eta-weak-pseudo-Hermiticity generators and radially symmetric Hamiltonians

A class of spherically symmetric non-Hermitian Hamiltonians and their \eta-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 \eta-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.Comment: 14 pages, content revised/regularized to cover 1D and 3D case

Complex Lagrangians and phantom cosmology

Motivated by the generalization of quantum theory for the case of non-Hermitian Hamiltonians with PT symmetry, we show how a classical cosmological model describes a smooth transition from ordinary dark energy to the phantom one. The model is based on a classical complex Lagrangian of a scalar field. Specific symmetry properties analogous to PT in non-Hermitian quantum mechanics lead to purely real equation of motion.Comment: 11 pages, to be published in J.Phys.A, refs. adde