Periodic Dirac Research Articles

Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta Potential

The solution of a quantum periodic potential in solid state physics is relevant because one can understand how electrons behave in a corresponding crystal. In this paper, the analytical solution of the energy formulation for a one-dimensional periodic potential that meets certain boundary conditions is developed in a didactic and detailed way. In turn, the group speed and effective mass are also calculated from the transcendental energy equation of a potential V=V(x). From this, a comparison is made with periodic potentials with known analytical solutions, such as the Dirac delta, as well as rectangular and triangular potentials. Finally, some limits are proposed in which these periodic potentials of the form V=V(x) approach the periodic Dirac delta potential of positive intensity. Therefore, the calculations described in this paper can be used as the basis for more-complex one-dimensional potentials and related simulations.

INTEGRATION OF A NONLINEAR HIROTA TYPE EQUATION WITH ADDITIONAL TERMS

In this paper, the inverse spectral problem method is used to integrate a nonlinear Hirota-type equation with additional terms in the class of periodic functions. The evolution of the spectral data of the periodic Dirac operator is introduced, the coefficient of Dirac operator is a solution to the nonlinear Hirota type equation with additional terms. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of six times continuously differentiable periodic infinite-zone functions is proved.

Cauchy Problem for the Nonlinear Liouville Equation in the Class of Periodic Infinite-Gap Functions

The inverse spectral problem method is used to integrate the nonlinear Liouville equation in the class of periodic infinite-gap functions. The evolution of the spectral data of the periodic Dirac operator whose coefficient is a solution of the nonlinear Liouville equation is introduced. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent function series constructed by solving the Dubrovin system of equations and using the first trace formula satisfies the Liouville equation.

Cauchy problem for the nonlinear Hirota equation in the class of periodic infinite-zone functions

In this paper, the method of inverse spectral problem is used to integrate the nonlinear Hirota equation in the class of periodic infinite-zone functions. An evolution of the spectral data of the periodic Dirac operator is introduced, where the coefficient of this operator is the solution of the nonlinear Hirota equation. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five times continuously differentiable periodic infinite-zone functions is shown. In addition, it is proved that if the initial function is a π \pi -periodic real-analytic function, then the solution of the Cauchy problem for the Hirota equation is also a real-analytic function in the variable x x ; and if the number π / 2 \pi /2 is a period (antiperiod) of the initial function, then the number π / 2 \pi /2 is a period (antiperiod) in the variable x x of the solution of the Cauchy problems for the Hirota equation.

Analysis of Finite Solution Spaces of Second-Order ODE with Dirac Delta Periodic Forcing

Second-order Ordinary Differential Equations (ODEs) with discontinuous forcing have numerous applications in engineering and computational sciences. The analysis of the solution spaces of non-homogeneous ODEs is difficult due to the complexities in multidimensional systems, with multiple discontinuous variables present in forcing functions. Numerical solutions are often prone to failures in the presence of discontinuities. Algebraic decompositions are employed for analysis in such cases, assuming that regularities exist, operators are present in Banach (solution) spaces, and there is finite measurability. This paper proposes a generalized, finite-dimensional algebraic analysis of the solution spaces of second-order ODEs equipped with periodic Dirac delta forcing. The proposed algebraic analysis establishes the conditions for the convergence of responses within the solution spaces without requiring relative smoothness of the forcing functions. The Lipschitz regularizations and Lebesgue measurability are not considered as preconditions maintaining generality. The analysis shows that smooth and locally finite responses can be admitted in an exponentially stable solution space. The numerical analysis of the solution spaces is computed based on combinatorial changes in coefficients. It exhibits a set of locally uniform responses in the solution spaces. In contrast, the global response profiles show localized as well as oriented instabilities at specific neighborhoods in the solution spaces. Furthermore, the bands of the expansions–contractions of the stable response profiles are observable within the solution spaces depending upon the values of the coefficients and time intervals. The application aspects and distinguishing properties of the proposed approaches are outlined in brief.

Periodic Dirac operator with dislocation

We consider Dirac operators with dislocation potentials on the line. The dislocation potential is a periodic potential for x < 0 and the same potential but shifted by t ∈ R for x > 0 . Its spectrum has an absolutely continuous part (the union of bands separated by gaps) plus at most two eigenvalues in each gap. Its resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface. We prove that it has only two simple poles on each open gap: eigenvalues or resonances. These poles are called states and there are no other poles. We prove: 1) states are continuous functions of t , and we obtain their local asymptotics; 2) for each t states in the gap are distinct; 3) states can be monotone or non-monotone functions of t ; 4) we construct examples of operators with different types of states in gaps.

Simplified Harmonic Rejection Mixer Analysis and Design Based on a Filtered Periodic Impulse Model

Hard-switched mixers have attractive linearity and noise properties, but are plagued by spurious responses due to the harmonic content of square-wave local oscillator (LO) signals. Harmonic rejection mixers (HRMs) reject (some of) these responses and can be constructed combining polyphase mixing and amplitude weighting. This brief presents a generalized model for the analysis of the harmonic rejection (HR) of such HRMs, based on circular convolution. We show that the effective LO signal can be modeled as a periodic Dirac impulse filtered by a time-discrete filter. For a multi-stage HR system, this filter consists of multiple stages as well, and the coefficients of each stage can be found simply by inspection. The total HR is shown to be the sum (in dB) of the rejection of the individual filtering stages, highly simplifying analysis and design of HRMs.

A simple real-space scheme for periodic Dirac operators

We address in this work the question of the discretization of two-dimensional periodic Dirac Hamiltonians. Standard finite differences methods on rectangular grids are plagued with the so-called Fermion doubling problem, which creates spurious unphysical modes. The classical way around the difficulty used in the physics community is to work in the Fourier space, with the inconvenience of having to compute the Fourier decomposition of the coefficients in the Hamiltonian and related convolutions. We propose in this work a simple real-space method immune to the Fermion doubling problem and applicable to all two-dimensional periodic lattices. The method is based on spectral differentiation techniques. We apply our numerical scheme to the study of flat bands in graphene subject to periodic magnetic fields and in twisted bilayer graphene.

Sensing improvement of 1D photonic crystal sensors by hybridization of defect and Bloch surface modes

There are enormous efforts to make new optical sensors based on photonic crystals with improved features. In this paper, we utilize the coupling behavior of two different localized modes of a one-dimensional photonic crystal to promote the sensing performance of sensors based on Bloch surface waves. Defect modes are appeared by embedding a periodic Dirac semimetal material-Dielectric layers into a one-dimensional photonic crystal. The dispersion relations of surface and the defect modes are obtained using the transfer matrix method. The hybridization of the defect and surface modes are examined and optimized by the period number and also the position of the defect layers, physical parameters of Dirac semimetal and the distance of defect layers from the prism. The sensing characteristics are studied under the Kretschmann configuration with the reflectance spectrum. The Bloch surface waves are excited in Kretschmann configuration and the sensing characteristics such as the angular sensitivity and the figure of merits are investigated by the reflection spectrum changes. The results reveal that foregoing parameters have great influences on the sensor performances. Besides, the results are compared with those of the structures with ordinary periodic dielectric- dielectric layers (no Dirac semimetal). Considerable improvement of sensor performance is achieved in these structures and consequently, our findings can be present promising applications of Bloch surface wave based sensors.

A relativistic one dimensional band model with position dependent mass

In this note a one-dimensional band model is proposed based on a periodic Dirac comb having a periodic mass distribution m(x). The mass function is represented as a Hermitian, non-local separable operator. Two specific cases–a constant mass model and a sinusoidal mass model–are examined. The lowest electron and positron bands for the constant mass case are similar to those for the standard relativistic Kronig-Penney model, suggesting that non-locality has little influence. The results for the sinusoidal case are consistent with the expectation that at low wavenumber an electron “feels” it has an average constant mass, but at high wave number, the particle “sees” the periodic mass variation and the band is distorted.

Topology and exceptional points of massive Dirac models with generic non-Hermitian perturbations

Recently, there has been a lot of activity in the research field of topological non-Hermitian physics, partly driven by fundamental interests and partly driven by applications in photonics. However, despite these activities, a general classification and characterization of non-Hermitian Dirac models that describe the experimental systems is missing. Here, we present a systematic investigation of massive Dirac models on periodic lattices, perturbed by general non-Hermitian terms. We find that there are three different types of non-Hermitian terms. For each case we determine the bulk exceptional points, the boundary modes, and the band topology. Our findings serve as guiding principles for the design of applications, for example, in photonic lattices. For instance, periodic Dirac systems with non-Hermitian mass terms can be used as topological lasers. Periodic Dirac systems with non-Hermitian anti-commuting terms, on the other hand, exhibit exceptional points at the surface, whose non-trivial topology could be utilized for optical devices.

Solutions of nonlinear periodic Dirac equations with periodic potentials

This paper is concerned with the nonlinear Dirac equation $ -i\sum_{k = 1}^{3}\alpha_{k}\partial_{k}u + [V(x)+a]\beta u + \omega u = f(x, u) $ in $ \mathbb{R}^3 $, where $ V(x) $ and $ f(x, u) $ are periodic in $ x $, $ f(x, u) $ is asymptotically linear and superlinear as $ |u|\rightarrow \infty $. Under weaker assumptions on $ f $, we obtain the existence of one nontrivial solution for the above equation.

The use of the generalized sinc-Gaussian sampling for numerically computing eigenvalues of periodic Dirac system

The generalized sinc-Gaussian sampling operator is established by Asharabi(2016) to approximate two classes of analytic functions. In this paper, weuse this operator to construct a new sampling method to approximate theeigenvalues of the periodic (semi-periodic) Dirac system of differentialequations problem. The convergence rate of this method is of exponentialtype, i.e., $\\mathrm{e}^{-\\alpha_{r} N}/\\sqrt{N}$,$\\alpha_{r}=\\left((r+1)\\pi-\\sigma h\\right)/2$. The sinc-Gaussian andHermite-Gauss methods are special cases of this method.We estimate the amplitude error associated to this operator, which gives us the possibility to establish the error analysis of this method.Various illustrative examples are presented and they show a good agreement with our theoretical analysis.

Uncertainty relations for characteristic functions

We present the uncertainty relation for the characteristic functions (ChUR) of the quantum mechanical position and momentum probability distributions. This inequality is more general than the Heisenberg Uncertainty Relation, and is saturated in two extremal cases for wavefunctions described by periodic Dirac combs. We further discuss a broad spectrum of applications of the ChUR, in particular, we constrain quantum optical measurements involving general detection apertures and provide the uncertainty relation that is relevant for Loop Quantum Cosmology. A method to measure the characteristic function directly using an auxiliary qubit is also briefly discussed.

Threshold conditions, energy spectrum and bands generated by locally periodic Dirac comb potentials

We derive expressions for polynomials governing the threshold conditions for different types of locally periodic Dirac comb potentials comprising of attractive and combination of attractive and repulsive delta potential terms confined symmetrically inside a one dimensional box of fixed length. The roots of these polynomials specify the conditions on the potential parameters in order to generate threshold energy bound states. The mathematical and numerical methods used by us were first formulated in our earlier works and it is also very briefly summarized in this paper. We report a number of mathematical results pertaining to the threshold conditions and these are useful in controlling the number of negative energy states as desired. We further demonstrate the correlation between the distribution of roots of these polynomials and negative energy eigenvalues. Using these results as basis, we investigate the energy bands in the positive energy spectrum for the above specified Dirac comb potentials and also for the corresponding repulsive case. In the case of attractive Dirac comb the base energy of the each band excluding the first band coincides with specific eigenvalue of the confining box whereas in the repulsive case it coincides with the band top. We deduce systematic correlation between band gaps, band spreads and box eigenvalues and explain the physical reason for the vanishing of band pattern at higher energies. In the case of Dirac comb comprising of orderly arranged attractive and repulsive delta potentials, specific box eigenvalues occur in the middle of each band excluding the first band. From our study we find that by controlling the number and strength parameters of delta terms in the Dirac comb and the size of confining box it is possible to generate desired types of band formations. We believe the results from our systematic analysis are useful and relevant in the study of various one dimensional systems of physical interest in areas like nanoscience.

Logarithmic Sobolev inequalities and spectral concentration for the cubic Schrödinger equation

The nonlinear Schrödinger equation , , arises from a Hamiltonian on infinite-dimensional phase space . For , Bourgain (Comm. Math. Phys. 166 (1994), 1–26) has shown that there exists a Gibbs measure on balls in phase space such that the Cauchy problem for is well posed on the support of , and that is invariant under the flow. This paper shows that satisfies a logarithmic Sobolev inequality (LSI) for the focusing case and on for all N>0; also satisfies a restricted LSI for on compact subsets of determined by Hölder norms. Hence for p = 4, the spectral data of the periodic Dirac operator in with random potential subject to are concentrated near to their mean values. The paper concludes with a similar result for the spectral data of Hill's equation when the potential is random and subject to the Gibbs measure of Korteweg–de Vries.

Homogenization of the two-dimensional periodic Dirac operator

We consider the homogenization problem for the two-dimensional periodic Dirac operator with magnetic potential of two types: singular and nonsingular. Based on the operator-theoretic approach due to M. Birman and T. Suslina, we approximate the resolvent in the space of bounded operators acting in $ {L_2}\left( {{{\mathbb{R}}^2};{{\mathbb{C}}^2}} \right) $ with order-sharp error estimates. Bibliography: 4 titles.

Equicontinuous Delone Dynamical Systems

AbstractWe characterize equicontinuous Delone dynamical systems as those coming from Delone sets with strongly almost periodic Dirac combs. Within the class of systems with finite local complexity, the only equicontinuous systems are then shown to be the crystallographic ones. On the other hand, within the class without finite local complexity, we exhibit examples of equicontinuous minimal Delone dynamical systems that are not crystallographic. Our results solve the problem posed by Lagarias as to whether a Delone set whose Dirac comb is strongly almost periodic must be crystallographic.

Instability of cnoidal‐peak for the NLS‐δ equation

AbstractWe study the existence and stability of standing waves for the periodic cubic nonlinear Schrödinger equation with a point defect determined by the periodic Dirac distribution at the origin. We show that this model admits a smooth curve of periodic‐peak standing wave solutions with a profile determined by the Jacobi elliptic function of cnoidal type. Via a perturbation method and continuation argument, we obtain that in the repulsive defect, the cnoidal‐peak standing wave solutions are unstable in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^1_{per}$\end{document} with respect to perturbations which have the same period as the wave itself. Global well‐posedness is verified for the Cauchy problem in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^1_{per}$\end{document} .

Tumour–host dynamics under radiotherapy

Tumour–host interaction is modelled by the Lotka–Volterra equations. Qualitative analysis and simulations show that this model reproduces all known states of development for tumours. Radiotherapy effect is introduced into the model by means of the linear-quadratic model and the periodic Dirac delta function. The evolution of the system under the action of radiotherapy is simulated and parameter space is obtained, from which certain threshold of effectiveness values for the frequency and applied doses are derived. A two-dimensional logistic map is derived from the modified Lotka–Volterra model and used to simulate the effectiveness of radiotherapy in different regimens of tumour development. The results show the possibility of achieving a successful treatment in each individual case by employing the correct therapeutic strategy.