One-dimensional Dirac Equation Research Articles

Reflectionless potentials for the one-dimensional Dirac equation: Pseudoscalar potentials

For the Dirac equation in one dimension, a method for constructing reflectionless pseudoscalar potentials is presented. This is a relativistic generalization of Kay and Moses's classic method [J. Appl. Phys. 27, 1503 (1956)] that deals with a similar problem for the Schr\"odinger equation. This can also be viewed as an inverse scattering problem. For a set of assumed energy levels, the method furnishes a family of potentials, each of which reproduces the energy levels. These potentials are all reflectionless.

A one-dimensional relativistic screened Coulomb potential

We propose a screened Coulomb potential leading to an exactly solvable one-dimensional Dirac equation. Unlike the one-dimensional Coulomb potential, the screened potential can support truly bound states because the Klein paradox is absent, provided that the potential does not dive into the negative-energy continuum. The δ-function limit of the potential is considered in detail. In the conclusions we discuss possible applications of our results in different physical contexts.

SPECTRAL PROPERTIES OF DERIVATIVE OPERATORS IN THE BASIS-SPLINE COLLOCATION METHOD

We discuss the basis-spline collocation method for the lattice solution of boundary-value differential equations, drawing particular attention to the difference between lattice and continuous collocation methods. Spectral properties of the basis-spline lattice representation of the first and second spatial derivatives are studied for the case of periodic boundary conditions with homogeneous lattice spacing and compared to spectra obtained using traditional finite-difference schemes. Basis-spline representations are shown to give excellent resolution on small-length scales and to satisfy the chain rule with good fidelity for the lattice-derivative operators using high-order splines. Application to the one-dimensional Dirac equation shows that very high-order spline representations of the Hamiltonian on odd lattices avoid the notorious spectral-doubling problem.

Supertransparent potentials for the Dirac equation

An extension of the classical Darboux transformations is applied to the one-dimensional Dirac equation in order to construct von Neumann-Wigner potentials allowing embedded eigenvalues. These potentials lead to a novel type of scattering problem with a trivial S-matrix composed of vanishing reflection coefficients and a trivial transmission coefficient. Related topics like the underlying symmetry of the Dirac equation and the connection with positon solutions of nonlinear evolution equations are discussed.

ON SPHALERON DEFORMATIONS INDUCED BY YUKAWA INTERACTIONS

Due to the presence of the chiral anomaly sphalerons, half a number of Chern-Simons (CS=1/2) are the only static configurations that allow a fermion level crossing in the two-dimensional Abelian-Higgs model with massless fermions, i.e. in the absence of Yukawa interactions. In the presence of fermion-Higgs interactions we demonstrate the existence of zero energy solutions to the one-dimensional Dirac equation at deformed sphalerons with CS ≠ 1/2. Induced level crossing due to Yukawa interactions illustrates a non-trivial generalization of the Atiyah-Patodi-Singer index theorem and of the equivalence between parity anomaly in odd and chiral anomaly in even dimensions. We discuss a subtle manifestation of this effect in the standard electroweak theory at finite temperatures.

A relativistic equation for a slowly varying potential

A relativistic equation is derived for a slowly varying potential by suitably approximating the one-dimensional Dirac equation. This equation is shown to be akin to the Schrodinger equation with an effective potential and effective eigenvalues. An iterative procedure for solving this equation is indicated. As an application, the relativistic treatment of the Mathieu potential on the basis of this equation is considered and results are compared with those obtained by solving the exact one-dimensional Dirac equation. These results are likely to take adequate account of the relativistic effects on electrons near Fermi levels in metals.

Localized solutions of one-dimensional nonlinear Dirac equations with point interaction potentials

The authors find the exact localized solutions of a class of nonlinear Dirac equations perturbed by a point interaction potential, that is, any sharply peaked potential approaching the delta -function limit. A detailed analysis of the existence conditions of these localized solutions is carried out. They show that their results agree with non-relativistic predictions when both the self-coupling and the point interaction potential are weak. Lower bounds to the size of the localized solutions are presented.

Boundary conditions across a delta -function potential in the one-dimensional Dirac equation.

We have developed a unified perspective for boundary conditions concerned with (I) a direct relativistic treatment of a \ensuremath{\delta}-function potential and (II) the \ensuremath{\delta}-function limit of solutions for relativistic treatment of peaked potential. We have presented a critical discussion of our findings, especially in the context of (i) certain conditions related very importantly to this unified perspective and (ii) the difficulty in making a choice between the approaches (I) and (II) on the basis of mathematical analyses alone.

Dirac equations for MHD waves: Hamiltonian spectra and supersymmetry

The propagation of MHD waves in a one-dimensional atmosphere is considered. For Alfven and acoustic waves the evolution equations are transformed into one-dimensional Dirac equations with a supersymmetric structure. For the case of an isothermal and hydrostatic atmosphere, rigorous spectral resolutions for the MHD Dirac Hamiltonians are given.

Inversion of transmission for the one-dimensional Dirac equation

It is shown that the transmission coefficient for the Dirac equation does not depend on the direction in which the barrier is traversed in one-dimensional quantum mechanics. As an explicit example, the case of two consecutive square barriers of unequal height and width are considered.

Transparent potential for the one-dimensional Dirac equation.

For the Dirac equation in one dimension, a transparent potential is constructed such that the transmission probability is unity at all energies. The potential is related to a soliton-type solution of the nonlinear Dirac equation with a specific type of nonlinearity that exhibits ``supersymmetry.''

A simple numerical method for the determination of relativistic one-dimensional band structures

A discretized form of the one-dimensional Dirac equation is solved to study relativistic band structures. Properties of periodic continued fractions are used to find the dispersion relation inside allowed bands. The Mathieu potential is considered and it is found that relativistic effects cause a reduction of the width of allowed bands and gaps.

Transmission properties of random point scatterers for waves with two-band dispersion law

The scattering problem is considered for the one-dimensional Dirac equation whose potential is a system of randomly distributed point scatterers. Types of such scatterers are described. The probability densities for the transmission and transformation disbalance coefficients are determined in the high-energy region and various averaged characteristic are calculated.

Quark tunneling in a one-dimensional nuclear model.

A general result for the eigenenergies of the one-dimensional Dirac equation for an arbitrary array of delta-function potentials of either Lorentz scalar or vector type is developed. This result is then used to discuss a finite one-dimensional model of the nucleus from which the energy bands of the constituent quarks may be obtained. The resulting energy eigenvalues are compared with those obtained previously via a similar model based on a Klein-Gordon equation. In contrast to this alternative treatment a system consisting of massless quarks confined to a regular periodic one-dimensional nucleus is found to be unbound. In the near ultra-relativistic limit binding to first order in the mass is found, thus conforming unexpectedly with the nonrelativistic limit. The possible surface states are examined and it is found that the so-called Dirac surface states are not expected for potentials of either Lorentz character. The existence and effect of quark surface states is also discussed in terms of our nuclear model. Finally the scattering coefficients for a relativistic particle in an arbitrary array of delta-function potentials are determined and their implications with regard to quark tunneling in nuclei are discussed.

Supercritical effects and the δ potential

The authors present a discussion of the occurrence of supercritical effects induced by an attractive δ potential. A detailed analysis of the one-dimensional Dirac equation in the presence of such a potential shows that the ground state never dives into the Dirac sea and therefore the supercritical effects are absent. In the three-dimensional spherically symmetric δ-potential case, in contrast, they find supercritical effects. This situation represents an interesting problem in relativistic quantum mechanics because the δ potential induces discontinuities in the wavefunction, in contrast to the well known Schrodinger case where only the first derivative of the wavefunction becomes discontinuous.

Low-energy scattering and Levinson's theorem for a one-dimensional Dirac equation

For a Dirac system on the line, the scattering matrix is defined in terms of Lippmann-Schwinger type solutions, which are also used to express an eigenfunction expansion. The S-matrix is shown to be continuous for potential functions whose first moments exist. Special attention is focused on the continuity of the S-matrix at the endpoints of the spectral gap (-c,c). Levinson's theorem follows as a corollary to this analysis. The author also obtains the leading-order behaviour for some specific power-law potential functions.

Bound states and confining properties of relativistic point interaction potentials

Bound states of the one-dimensional Dirac equation for vector plus Lorentz scalar point interaction potentials have been obtained. Confining properties of these potentials are briefly discussed.

An extension of the Saxon-Hutner theorem in the relativistic domain

Abstract It is shown that to any real localized potential of the one-dimensional Dirac equation one may juxtapose a vector of special type in the three-dimensional complex vector space. A sufficient condition for the validity of the Saxon-Hutner theorem is given in terms of these vectors. Subramanian and Bhagwat's relativistic generalization of Luttinger's proof is confirmed and the “converse” theorem is briefly discussed.

Particle scattering at the Planck scale and the Aichelburg-Sexl geometry

At energies of the order or larger than the Planck mass, the curved space-time geometry created by the particles dominates their collision process. The so-called Aichelburg-Sexl (AS) metric is relevant in this problem. We study the exact quantum particle scattering by this geometry in D space-time dimensions. The Klein-Gordon equation is ill defined and a regularization procedure is needed to give a sense to it (this does not happen for string equations). We find the troublesome result that the exact solution depends on the regularization chosen. Continuous regularization yields a scattering phase shift [ ϕ cont] that agrees with that found by 't Hooft in D=4. Discrete (lattice) regularization yields in the continuum limit a phase-shift ϕ = 2arctang[ ϕ cont]. (Actually, the same lattice results hold for the one-dimensional Dirac equation with a Dirac-δ potential and for fermionic two-dimensional field-theoretical models.) In the AS metric, we find that the scattering amplitude [ f lattice], obtained from lattice regularization, exhibits cuts in both s and t variables, and can be expressed as a coherent superposition of relativistic coulombian amplitudes [ f cont obtained from continuous regularization exhibits poles]. For small s, both f cont and f lattice= iGs( πt) −1 (i.e. the one-graviton exchange amplitude) but for small t, f lattice =i[Gsπt ln 2 (− 1 4 t)]−1 (f cont =iGs(πt) −1 for both small s and t). We compute and analyze the partial-wave amplitudes. We also compare both amplitudes in the intermediate t region and discuss their connection with the eikonal (and improved eikonal) approximation.

One-dimensional dirac equation approach to the effective mass concept in diatomic crystals

A relativistic generalization of the effective-mass concept is worked out on the basis of simplest one-dimensional model of diatomic crystals and in the framework of the Dirac equation approach. The numerical analysis performed leads to reasonable values for both, the energy gaps and the effective masses of the carriers. The relativistic corrections thus obtained bring on a firm basis the justification of simpler calculations in the framework of the Schrodinger equation. The need of more powerful approaches beyond the Dirac equation is outlined, especially for crystals of very heavy atoms (Z > 68). Eine relativistische Verallgemeinerung des Grundbegriffs der effektiven Masse wird auf Grund von einfachsten eindimensionalen Modellen zweiatomarer Kristalle und mit Hilfe der Diracgleichung entwickelt. Fur die verbotenen Zonen und fur die effektiven Massen werden dabei annehmbare Werte abgeleitet. Die geringen relativistischen Korrekturen beweisen, das die einfachere Berechnung im Rahmen der Schrodingergleichung berechtigt sind. Fur Kristalle aus schwereren Atomen (Z > 68) sind jedoch andere verfeinerte Methoden notwendig, da hier die Diracgleichung (auch dreidimensional) bestimmte Unvollkommenheiten zeigt.