Complex Conjugate Pairs Research Articles

All-real spectra in optical systems with arbitrary gain-and-loss distributions

A method for constructing optical potentials with an arbitrary distribution of gain and loss and completely real spectrum is presented. For each arbitrary distribution of gain and loss, several classes of refractive-index profiles with freely tunable parameters are obtained such that the resulting complex potentials, although being non-parity-time-symmetric in general, still feature all-real spectra for a wide range of tuning parameters. When these refractive indices are tuned below certain thresholds, phase transition can occur, where complex-conjugate pairs of eigenvalues appear in the spectrum. These non-parity-time-symmetric complex potentials generalize the concept of parity-time-symmetric potentials to allow for more flexible gain and loss distributions while still maintaining all-real spectra and the phenomenon of phase transition.

Peak current mode-controlled pulse width-modulated active clamp forward converter with centre tap transformer

This paper presents the control to output transfer function, principles and performance of peak current mode-controlled active clamp forward converter with centre-tap transformer. There is a clamp circuit comprising of one auxiliary switch and one clamp capacitor, which is used in the forward converter to recycle the energy stored in the magnetising inductor. Due to presence of clamp circuit, the active clamp forward converter with centre tap transformer exhibits fourth-order power stage dynamics. The current mode control is implemented to enhance the performance of the converter by sensing the main switch current. It is demonstrated that current loop gain splits not only the complex-conjugate pole of the output filter but also the complex-conjugate pole pair formed by clamp circuit into real poles. Different transfer functions are derived using state space average model. The peak current mode control is implemented experimentally using controller IC, UCC2891D, and the simulation results are verified with the experimental setup.

FIR Filter Design Using Optimal Factoring: A Walkthrough and Summary of Benefits

Decomposing an FIR filter into a cascade of blocks, each representing an optimal factor of H(z), the filter?s transfer function, can start with the complex-conjugate pairs of zeros of H(z) and then creating optimal pairings of these zero-pairs. This leads to a general method for designing FIR filters having minimal hardware complexity. We illustrate this concept, further techniques, benefits, and performance improvements in this paper, highlighting the practical features of the design method. It is also shown that the resulting structure can achieve lower hardware complexity, when compared with the prior art.

Extension of the CPT theorem to non-Hermitian Hamiltonians and unstable states

We extend the CPT theorem to quantum field theories with non-Hermitian Hamiltonians and unstable states. Our derivation is a quite minimal one as it requires only the time-independent evolution of scalar products, invariance under complex Lorentz transformations, and a non-standard but nonetheless perfectly legitimate interpretation of charge conjugation as an antilinear operator. The first of these requirements does not force the Hamiltonian to be Hermitian. Rather, it forces its eigenvalues to either be real or to appear in complex conjugate pairs, forces the eigenvectors of such conjugate pairs to be conjugates of each other, and forces the Hamiltonian to admit of an antilinear symmetry. The latter two requirements then force this antilinear symmetry to be CPT, while forcing the Hamiltonian to be real rather than Hermitian. Our work justifies the use of the CPT theorem in establishing the equality of the lifetimes of unstable particles that are charge conjugates of each other. We show that the Euclidean time path integrals of a CPT-symmetric theory must always be real. In the quantum-mechanical limit the key results of the PT symmetry program of Bender and collaborators are recovered, with the C-operator of the PT symmetry program being identified with the linear component of the charge conjugation operator.

The Spectrum of a Nonlocal Difference Operator with Variable Coefficients

The spectrum of a non-symmetric matrix A is studied, which resulting from difference approximation of the differential operators of the second order with nonlocal boundary conditions and variable coefficients. Such matrices can be both simple and double real eigenvalue and a complex conjugate pairs of eigenvalues. The original matrix A is mapped to the so-called associated matrix L --- symmetric tridiagonal matrix, little differing from A . It is known that all eigenvalues of the matrix L are real and different. They can be found by the bisection method using a sequence of the Sturm's polynomials. In the paper on the basis of numerous examples are compared the properties of the spectra of matrices L and A . The method offers of approximate calculating of eigenvalues of the matrix A .

A Non Quasi Exact Solvable Eigenvalue Problem with PT-Invariant Non-Hermitian Complex Potential

It is well known that the quasi-exact solvable eigenvalues of the Schrödinger equation with potential V(x)=-(ξcosh2x-iM)2 is real for PT-invariant non-Hermitian potential in case the parameter M is odd integer and complex conjugate pairs when M is even integer. In this work the Asymptotic Iteration Method (AIM) were used to solve this problem for M odd and even integer, and for any non-integer M values.

Pole Placement in Single-Input Linear Descriptor Systems

A double-step algorithm is presented to solve the pole placement problem for descriptor single-input linear systems using state feedback. The algorithm places two finite poles (eigenvalues) at a time so that complex conjugate pairs of poles are placed using only real arithmetic. Possible infinite poles of the open loop system are replaced by finite poles, therefore making the closed loop system either causal (discrete time) or free of impulsive behavior (continuous time). The numerical stability of the algorithm is proved by doing a backward rounding error analysis. It is worth pointing out that the algorithm is closely related to the double-step QZ algorithm for the generalized eigenproblem.

Exciting flavored bound states

We study ground and radial excitations of flavor singlet and flavored pseudoscalar mesons within the framework of the rainbow-ladder truncation using an infrared massive and finite interaction in agreement with recent results for the gluon-dressing function from lattice QCD and Dyson-Schwinger equations. Whereas the ground-state masses and decay constants of the light mesons as well as charmonia are well described, we confirm previous observations that this truncation is inadequate to provide realistic predictions for the spectrum of excited and exotic states. Moreover, we find a complex conjugate pair of eigenvalues for the excited ${D}_{(s)}$ mesons, which indicates a non-Hermiticity of the interaction kernel in the case of heavy-light systems and the present truncation. Nevertheless, limiting ourselves to the leading contributions of the Bethe-Salpeter amplitudes, we find a reasonable description of the charmed ground states and their respective decay constants.

Unified Frequency-Dependent Split-Step FDTD Formulations Based on the Complex-Conjugate Pole-Residue Pairs Model

Unified and efficient frequency-dependent split-step finite difference time domain (SS-FDTD) formulations are presented. In the proposed formulations, the complex-conjugate pole-residue (CCPR) pairs model is adopted in the FDTD implementation of the domain's frequency-dependent relation with the minimal additional auxiliary variables. In addition, the formulations allow modeling different dispersive materials, such as Debye, Drude, and Lorentz, in the same manner. A numerical example is included to show the validity of the formulations.

A remark about the positivity problem of fourth order linear recurrence sequences

Consider a fourth order linear recurrence with integer coefficients whose characteristic polynomial has two distinct real and a complex conjugate pair of roots. A new proof showing that its positivity problem is decidable is given for the case where there is exactly one real root having the same absolute value as the two complex conjugate roots.

Hofstadter’s Cocoon

Hofstadter showed that the energy levels of electrons on a lattice plotted as a function of magnetic field form an beautiful structure now referred to as "Hofstadter's butterfly". We study a non-Hermitian continuation of Hofstadter's model; as the non-Hermiticity parameter $g$ increases past a sequence of critical values the eigenvalues successively go complex in a sequence of "double-pitchfork bifurcations" wherein pairs of real eigenvalues degenerate and then become complex conjugate pairs. The associated wavefunctions undergo a spontaneous symmetry breaking transition that we elucidate. Beyond the transition a plot of the real parts of the eigenvalues against magnetic field resembles the Hofstadter butterfly; a plot of the imaginary parts plotted against magnetic fields forms an intricate structure that we call the Hofstadter cocoon. The symmetries of the cocoon are described. Hatano and Nelson have studied a non-Hermitian continuation of the Anderson model of localization that has close parallels to the model studied here. The relationship of our work to that of Hatano and Nelson and to PT transitions studied in PT quantum mechanics is discussed.

A phenomenological approach to the equation of state of a unitary Fermi gas

We propose a phenomenological approach for the equation of state of a unitary Fermi gas. The universal equation of state is parametrized in terms of Fermi–Dirac integrals. This reproduces the experimental data over the accessible range of fugacity and normalized temperature, but cannot describe the superfluid phase transition found in the MIT experiment [Ku et al, Science 335, 563 (2012)]. The most sensitive data for compressibility and specific heat at phase transition can, however, be fitted by introducing into the grand partition function a pair of complex conjugate zeros lying in the complex fugacity plane slightly off the real axis.

THE FLOQUET METHOD FOR PT-SYMMETRIC PERIODIC POTENTIALS

By the general theory of PT-symmetric quantum systems, their energy levels are either real or occur in complex-conjugate pairs, which implies that the secular equation must be real. However, for periodic potentials it is by no means clear that the secular equation arising in the Floquet method is indeed real, since it involves two linearly independent solutions of the Schrödinger equation. In this brief note we elucidate how that reality can be established.

PT symmetry as a necessary and sufficient condition for unitary time evolution

While Hermiticity of a time-independent Hamiltonian leads to unitary time evolution, in and of itself, the requirement of Hermiticity is only sufficient for unitary time evolution. In this paper, we provide conditions that are both necessary and sufficient. We show that PT symmetry of a time-independent Hamiltonian, or equivalently, reality of the secular equation that determines its eigenvalues, is both necessary and sufficient for unitary time evolution. For any PT-symmetric Hamiltonian H, there always exists an operator V that relates H to its Hermitian adjoint according to VHV(-1)=H(†). When the energy spectrum of H is complete, Hilbert space norms <ψ(1)|V|ψ(2)> constructed with this V are always preserved in time. With the energy eigenvalues of a real secular equation being either real or appearing in complex conjugate pairs, we thus establish the unitarity of time evolution in both cases. We also establish the unitarity of time evolution for Hamiltonians whose energy spectra are not complete. We show that when the energy eigenvalues of a Hamiltonian are real and complete, the operator V is a positive Hermitian operator, which has an associated square root operator that can be used to bring the Hamiltonian to a Hermitian form. We show that systems with PT-symmetric Hamiltonians obey causality. We note that Hermitian theories are ordinarily associated with a path integral quantization prescription in which the path integral measure is real, while in contrast, non-Hermitian but PT-symmetric theories are ordinarily associated with path integrals in which the measure needs to be complex, but in which the Euclidean time continuation of the path integral is nonetheless real. Just as the second-order Klein-Gordon theory is stabilized against transitions to negative frequencies because its Hamiltonian is positive-definite, through PT symmetry, the fourth-order derivative Pais-Uhlenbeck theory can equally be stabilized.

On the zeros of a partial theta function

The series θ(q,x):=∑j=0∞qj(j+1)/2xj converges for q∈[0,1), x∈R and defines a partial theta function. For any q∈(0,1) fixed it has infinitely many negative zeros. For countably many values q˜j of q said to form the spectrum of θ (where 0.3092493386…=q˜1<q˜2<⋯<1, limj→∞q˜j=1) the function θ(q,.) has a double zero which is the rightmost of its real zeros (the rest of them being simple). For q≠q˜j it has no multiple real zeros. For q∈(q˜N,q˜N+1) the function θ(q,.) has exactly N complex conjugate pairs of zeros counted with multiplicity (we set q˜0=0). If ξkl denote the zeros of ∂θl/∂xl(q,.) in the order of decreasing, then limk→∞ξklqk=−1 and limk→∞ξk+1l/ξkl=q.

Conserved correlation in [formula omitted]-symmetric systems: Scattering and bound states

For one-dimensional PT-symmetric systems, it is observed that the non-local product ψ∗(−x,t)ψ(x,t), obtained from the continuity equation can be interpreted as a conserved correlation function. This leads to physical conclusions regarding both discrete and continuum states of such systems. Asymptotic states are shown to have necessarily broken PT-symmetry, leading to modified scattering and transfer matrices. This yields restricted boundary conditions, e.g., incidence from both sides, analogous to that of the proposed PT CPA laser (Longhi, 2010) [4]. The interpretation of ‘left’ and ‘right’ states leads to a HermitianS-matrix, resulting in the non-conservation of the ‘flux’. This further satisfies a ‘duality’ condition, identical to the optical analogues (Paasschens et al., 1996) [17]. However, the non-local conserved scalar implements alternate boundary conditions in terms of ‘in’ and ‘out’ states, leading to the pseudo-Hermiticity condition in terms of the scattering matrix. Interestingly, when PT-symmetry is preserved, it leads to stationary states with real energy, naturally interpretable as bound states. The broken PT-symmetric phase is also captured by this correlation, with complex-conjugate pair of energies, interpreted as resonances.

A Modal Identification Algorithm Combining Blind Source Separation and State Space Realization

A modal identification algorithm is developed, combining techniques from Second Order Blind Source Separation (SOBSS) and State Space Realization (SSR) theory. In this hybrid algorithm, a set of correlation matrices is generated using time-shifted, analytic data and assembled into several Hankel matrices. Dissimilar left and right matrices are found, which diagonalize the set of nonhermetian Hankel matrices. The complex-valued modal matrix is obtained from this decomposition. The modal responses, modal auto-correlation functions and discrete-time plant matrix (in state space modal form) are subsequently identified. System eigenvalues are computed from the plant matrix to obtain the natural frequencies and modal fractions of critical damping. Joint Approximate Diagonalization (JAD) of the Hankel matrices enables the under determined (more modes than sensors) problem to be effectively treated without restrictions on the number of sensors required. Because the analytic signal is used, the redundant complex conjugate pairs are eliminated, reducing the system order (number of modes) to be identified half. This enables smaller Hankel matrix sizes and reduced computational effort. The modal auto-correlation functions provide an expedient means of screening out spurious computational modes or modes corresponding to noise sources, eliminating the need for a consistency diagram. In addition, the reduction in the number of modes enables the modal responses to be identified when there are at least as many sensors as independent (not including conjugate pairs) modes. A further benefit of the algorithm is that identification of dissimilar left and right diagonalizers preclude the need for windowing of the analytic data. The effectiveness of the new modal identification method is demonstrated using vibration data from a 6 DOF simulation, 4-story building simulation and the Heritage court tower building.

Pseudo-Hermitian quantum dynamics of tachyonic spin-1/2 particles

We investigate the spinor solutions, the spectrum and the symmetry properties of a matrix-valued wave equation whose plane-wave solutions satisfy the superluminal (tachyonic) dispersion relation , where E is the energy, is the spatial momentum and m is the mass of the particle. The equation reads (iγμ ∂μ − γ5 m)ψ = 0, where γ5 is the fifth current. The tachyonic equation is shown to be invariant and invariant. The tachyonic Hamiltonian breaks parity and is non-Hermitian but fulfils the pseudo-Hermitian property , where P is the parity matrix and is the full parity transformation. The energy eigenvalues and eigenvectors describe a continuous spectrum of plane-wave solutions (which correspond to real eigenvalues for ) and evanescent waves, which constitute resonances and anti-resonances with complex-conjugate pairs of resonance eigenvalues (for ). In view of additional algebraic properties of the Hamiltonian which supplement the pseudo-Hermiticity, the existence of a resonance energy eigenvalue E implies that E*, −E and −E* also constitute resonance energies of H5.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

Level spacings for weakly asymmetric real random matrices and application to two-color QCD with chemical potential

We consider antisymmetric perturbations of real symmetric matrices in the context of random matrix theory and two-color quantum chromodynamics. We investigate the level spacing distributions of eigenvalues that remain real or become complex conjugate pairs under the perturbation. We work out analytical surmises from small matrices and show that they describe the level spacings of large random matrices. As expected from symmetry arguments, these level spacings also apply to the overlap Dirac operator for two-color QCD with chemical potential.

Wentzel-Kramers-Brillouin analysis ofPT-symmetric Sturm-Liouville problems

In a previous paper it was shown that a one-turning-point WKB approximation gives an accurate picture of the spectrum of certain non-Hermitian PT-symmetric Hamiltonians on a finite interval with Dirichlet boundary conditions. Potentials to which this analysis applies include the linear potential $V=igx$ and the sinusoidal potential $V=ig\sin(\alpha x)$. However, the one-turning-point analysis fails to give the full structure of the spectrum for the cubic potential $V=igx^3$, and in particular it fails to reproduce the critical points at which two real eigenvalues merge and become a complex-conjugate pair. The present paper extends the method to cases where the WKB path goes through a {\it pair} of turning points. The extended method gives an extremely accurate approximation to the spectrum of $V=igx^3$, and more generally it works for potentials of the form $V=igx^{2N+1}$. When applied to potentials with half-integral powers of $x$, the method again works well for one sign of the coupling, namely that for which the turning points lie on the first sheet in the lower-half plane.