On quantum electrodynamics of two-particle bound states containing spinless particles

We develop here the general treatment arising from the Bethe-Salpeter equation for a two-particle bound system in which at least one of the particles is spinless. It is shown that a natural two-component formalism can be formulated for describing the propagators of scalar particles. This leads to a formulation of the Bethe-Salpeter equation in a form very reminiscent of the fermion-fermion case. It is also shown, that using this two-component formulation for spinless particles, the perturbation theory can be systematically developed in a manner similar to that of fermions. Quantum electrodynamics for scalar particles is then developed in the two component formalism, and the problem of bound states, in which one of the constituent particles is spinless, is examined by means of the means of the Bethe-Salpeter equation. For this case, the Bethe-Salpeter equation is cast into a form which is convenient to perform a Foldy-Woutyhuysen transformation which we carry out, keeping the lowest-order relativistic corrections to the nonrelativistic equation. The results are compared with the corresponding fermion-fermion case. It is shown, as might have been expected, that the only spin-independent terms that occur for the fermion-fermion system which do not occur for bound scalar particle cases, is the zitterbewegung contribution. The relevance of the above considerations for systems that are essentially bound by electromagnetic interactions, such as kaonic hydrogen, is discussed.     

On the Ladder Bethe-Salpeter Equation

The Bethe-Salpeter (BS) equation in the ladder approximation is studied within a scalar theory: two scalar fields (constituents) with mass m interacting via an exchange of a scalar field (tieon) with mass . The BS equation is written in the form of an integral equation in the configuration Euclidean x-space with the kernel which for stable bound states M < 2m is a self-adjoint positive operator. The solution of the BS equation is formulated as a variational problem. The nonrelativistic limit of the BS equation is considered. The role of so-called abnormal states is discussed.The analytical form of test functions for which the accuracy of calculations of bound-state masses is better than 1% (the comparison with available numerical calculations is done) is determined. These test functions make it possible to calculate analytically vertex functions describing the interaction of bound states with constituents.As a by-product a simple solution of the Wick-Cutkosky model for the case of massless bound states is demonstrated.     

On a relativistic scalar particle subject to a Coulomb-type potential given by Lorentz symmetry breaking effects

The behaviour of a relativistic scalar particle in a possible scenario that arises from the violation of the Lorentz symmetry is investigated. The background of the Lorentz symmetry violation is defined by a tensor field that governs the Lorentz symmetry violation out of the Standard Model Extension. Thereby, we show that a Coulomb-type potential can be induced by Lorentz symmetry breaking effects and bound states solutions to the Klein–Gordon equation can be obtained. Further, we discuss the effects of this Coulomb-type potential on the confinement of the relativistic scalar particle to a linear confining potential by showing that bound states solutions to the Klein–Gordon equation can also be achieved, and obtain a quantum effect characterized by the dependence of a parameter of the linear confining potential on the quantum numbers {n,l}$\{n,l\}$ of the system.     

Propagators for Scalar Bound States at Finite Temperature in an NJL Model

We re-examine physical causal propagators for scalar and pseudoscalar bound states at finite temperaturein a chiral Ut(1) x UR(1) NJL model, defined by four-point amputated fimctions subtracted through the gap equation,and prove that they are completely equivalent in the imaginary-time and real-time formalisms by separating carefiullythe imaginary part of the zero-temperature loop integral. It is shown that the same thermal transformation matrix ofthe matrix propagators for these bound states in the real-time formalism is precisely the one of the matrix propagatorfor an elementary scalar particle and this fact shows the similarity of thermodynamic property between a composite andelementary scalar particle. The retarded and advanced propagators for these bound states are also given explicitly fromthe imaginary-time formalism.     

具有一维Coulomb型对称势Dirac方程的精确解

在标量势大于矢量势的情况下,一维Dirac方程的束缚态能级是二重简并的.任意两个不同能量本征值的波函数和同一能量本征值的两个波函数都是相互正交的.对于纯标量场,存在零能量束缚态,存在分数电荷 关键词： Coulomb型对称势 Dirac方程 束缚态 分数电荷     

Mass hierarchy, mass gap and corrections to Newton’s law on thick branes with Poincaré symmetry

We consider a scalar thick brane configuration arising in a 5D theory of gravity coupled to a self-interacting scalar field in a Riemannian manifold. We start from known classical solutions of the corresponding field equations and elaborate on the physics of the transverse traceless modes of linear fluctuations of the classical background, which obey a Schrödinger-like equation. We further consider two special cases in which this equation can be solved analytically for any massive mode with $m^2\ge 0$ , in contrast with numerical approaches, allowing us to study in closed form the massive spectrum of Kaluza–Klein (KK) excitations and to analytically compute the corrections to Newton’s law in the thin brane limit. In the first case we consider a novel solution with a mass gap in the spectrum of KK fluctuations with two bound states—the massless 4D graviton free of tachyonic instabilities and a massive KK excitation—as well as a tower of continuous massive KK modes which obey a Legendre equation. The mass gap is defined by the inverse of the brane thickness, allowing us to get rid of the potentially dangerous multiplicity of arbitrarily light KK modes. It is shown that due to this lucky circumstance, the solution of the mass hierarchy problem is much simpler and transparent than in the thin Randall–Sundrum (RS) two-brane configuration. In the second case we present a smooth version of the RS model with a single massless bound state, which accounts for the 4D graviton, and a sector of continuous fluctuation modes with no mass gap, which obey a confluent Heun equation in the Ince limit. (The latter seems to have physical applications for the first time within braneworld models). For this solution the mass hierarchy problem is solved with positive branes as in the Lykken–Randall (LR) model and the model is completely free of naked singularities. We also show that the scalar–tensor system is stable under scalar perturbations with no scalar modes localized on the braneworld configuration.     

Quasibound states in semiconductor quantum well structures

We present a study on quasibound states in multiple quantum well structures using a finite element model (FEM). The FEM is implemented for solving the effective mass Schrödinger equation in arbitrary layered semiconductor nanostructures with an arbitrary applied potential. The model also includes nonparabolicity effects by using an energy dependent effective mass, where the resulting nonlinear eigenvalue problem was solved using an iterative approach. We focus on quasibound/continuum states above the barrier potential and show that such states can be determined using cyclic boundary conditions. This new method enables the determination of both bound and quasibound states simultaneously, making it more efficient than other methods where different boundary conditions have to be used in extracting the relevant states. Furthermore, the new method lifted the problem of quasibound state divergence commonly seen with many other methods of calculation. Hence enabling accurate determination of dipole matrix elements involving both bound and quasibound states. Such calculations are vital in the design of intersubband optoelectronic devices and reveal the interesting properties of quasibound states above the potential barriers.     

具有Hulthén的型势的Dirac方程的束缚态

本文给出,具有一维Hulthén型标量势和矢量势的Dirac方程的束缚态精确解。对于三维情况,在Hulthén型标量势和矢量势相等的条件下,给出Dirac方程的S波解。     

具有P?schl-Teller型标量势与矢量势的Klein-Gordon方程和Dirac方程的束缚态

给出了具P?schl-Teller型标量势与矢量势的Klein-Gordon方程和Dirac方程的s波束缚态解. 关键词： P?schl-Teller势 Klein-Gordon方程 Dirac方程 束缚态     

Problems with vector confinement in 4d QCD

It is shown that vector confinement does not support bound state spectrum in the 4d Dirac equation. The same property is confirmed in the heavy–light and light–light QCD systems. This situation is compared with the confinement in the 2d system, which is generated by the gluon exchange. Considering the existing theories of confinement, it is shown that both the field correlator approach and the dual superconductor model ensure the scalar confinement in contrast to the Gribov–Zwanziger model, where the confining Coulomb potential does not support bound states in the Dirac equation.     

库仑势加新环形势的相对论束缚态

在标量势等于矢量势的条件下，获得了库仑势加新环形势的Klein-Gordon方程和Dirac方程的束缚态的精确解. 对于Klein-Gordon方程，获得了精确的能谱方程和归一化的波函数. 对于Dirac方程，给出了精确的能谱方程和归一化的旋量波函数. 关键词： 库仑势加新环形势 束缚态 精确解     

On the Klein–Gordon oscillator subject to a Coulomb-type potential

By introducing the scalar potential as modification in the mass term of the Klein–Gordon equation, the influence of a Coulomb-type potential on the Klein–Gordon oscillator is investigated. Relativistic bound states solutions are achieved to both attractive and repulsive Coulomb-type potentials and the arising of a quantum effect characterized by the dependence of angular frequency of the Klein–Gordon oscillator on the quantum numbers of the system is shown.     

Solutions of the Dirac Equation for the Davidson Potential

In the present paper we solve the Dirac equation with Davidson potential by Nikiforov-Uvarov method. The Dirac Hamiltonian contains a scalar S and a vector V Davidson potentials. With equal scalar and vector potential, analytical solutions for bound states of the corresponding Dirac equations are found.     

Evidence against a scalar confining potential in QCD

We have written and solved numerically the Bethe-Salpeter equation for q- bound states, taking as kernel an instantaneous scalar confining potential. The resulting spectrum is inconsistent with the fact that light quark mesons lie on linearly rising Regge trajectories.     

Pseudospin symmetry for modified Rosen-Morse potential including a Pekeris-type approximation to the pseudo-centrifugal term

By a novel algebraic method we study the approximate solution to the Dirac equation with scalar and vector second P?schl-Teller potential carrying spin symmetry. The transcendental energy equation and spinor wave functions with arbitrary spin-orbit coupling quantum number k are presented. It is found that there exist only positive-energy bound states in the case of spin symmetry. Also, the energy eigenvalue approaches a constant when the potential parameter a \alpha goes to zero. The equally scalar and vector case is studied briefly.     

Solution of Schrödinger and related equations for irregular and composite regions

The Schrödinger equation with a potential is mathematically equivalent to the Helmholtz equation with a spatially variable propagation constant. A new method is presented for solving certain standard problems associated with these equations. In the Schrödinger language these are the ones in which a potential ν(|r|) acts inside an irregular (aspherical) boundary, where r has its origin inside the boundary. In terms of the Helmholtz equation, these include problems in which a region of constant index of refraction, but arbitrary shape, is embedded in a second uniform region with a different index. It is shown how bound state and scattering problems for such a potential (or region) can be treated in a way that avoids the usually intractable problem of matching solutions across the irregular boundary. The method requires, in general, the truncation of an infinite set of equations for partial wave amplitudes. The special case is discussed of a potential that becomes infinite throughout a region, so the wave amplitude must vanish inside the region (and, hence, on its boundary). For a long wave length this becomes a problem with the Laplace equation, and the general technique is illustrated by a calculation of the free charge on a perfectly conducting spheroid. The theory is extended from a single potential to an ensemble of such potentials, and in particular to an ensemble of potentials with spherical boundaries. In the special case that the potentials are arranged in a periodic lattice the formulas resemble those obtained by the KKR method, but are simpler in some ways. The method is extended to an ensemble of irregular potentials, and these results are shown to be applicable to the special case of an ensemble of finite range, but overlapping, spherical potentials.     

具有Kratzer型标量势与矢量势的Klein-Gordon方程和Dirac方程的束缚态

给出了具有Kratzer型标量势与矢量势的Klein-Gordon方程和Dirac方程的s波束缚态解.     

The spin symmetry for deformed generalized Pöschl-Teller potential

In the case of spin symmetry we solve the Dirac equation with scalar and vector deformed generalized Pöschl-Teller (DGPT) potential and obtain exact energy equation and spinor wave functions for s-wave bound states. We find that there are only positive energy states for bound states in the case of spin symmetry based on the strong regularity restriction condition λ<−η for the wave functions. The energy eigenvalue approaches a constant when the potential parameter α goes to zero. Two special cases such as generalized PT potential and standard PT potential are also briefly discussed.     

Absorption of a Massive Scalar Field by Wormhole Space-Times

In this paper we consider the problem of the test massive scalar field propagating in the background of a class of wormhole space-times. Basing on the quantum scattering theory, we analyze the Schrödinger-type scalar wave equation and compute transmission coefficients for arbitrary coupling of the field to the background geometry with the WKB approximation. We numerically investigate its absorption cross section and analyze them in the high frequency regime. We find that the absorption cross section oscillates about the geometric optical value and the limit of absorption cross section is uniform in the high frequency regime.     

Relativistic symmetry of position-dependent mass particle in Coulomb field including tensor interaction

The spatially-dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials including a tensor interaction under the spin and pseudospin symmetric limit. Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spin-orbit quantum number κ. Some numerical results are given too. The effect of the tensor interaction on the bound states is presented. It is shown that the tensor interaction removes the degeneracy between two states in the spin doublets. We also investigate the effects of the spatially-dependent mass on the bound states under the conditions of the spin symmetric limit in the absence of tensor interaction.