Exact Solutions of Klein-Gordon Equation with Exponential Scalar and Vector Potentials

We obtain the exact analytical solution of the Klein-Gordon equation for the exponential vector and scalar potentials by using the asymptotic iteration method. For the scalar potential greater than the vector potential case, the exact bound state energy eigenvalues and corresponding eigenfunctions are presented. The bound state eigenfunction solutions are obtained in terms of the confluent hypergeometric functions.     

Exact Solutions of the Klein–Gordon Equation with Hylleraas Potential

We present the exact solution of the Klein–Gordon with Hylleraas Potential using the Nikiforov–Uvarov method. We obtain explicitly the bound state energy eigenvalues and the corresponding eigen function for s-wave. The wave functions obtained are expressed in terms of Jacobi polynomials.     

Exact Solutions of the Klein–Gordon Equation for Spherically Asymmetrical Singular Oscillator

The relativistic Klein–Gordon equation with equal scalar and vector spherically asymmetrical singular oscillators is solved using the asymptotic iteration method. The energy eigenvalues equation and the corresponding wave functions are obtain explicitly. It was found that the asymptotic iteration method provides the closed-forms for the energy eigenvalues as well as the eigenfunctions. The non-relativistic limit ${c \rightarrow \infty}$ of the energy spectrum, where c is the speed of light, have also been discussed.     

Exact Solution of Klein–Gordon Equation for Hua Plus Modified Eckart Potentials

We report the exact s-wave solutions of the Klein–Gordon equation under equal scalar and vector the Hua plus modified Eckart potentials using the functional analysis method. The results, in special cases, yield the results of Morse, Hua, Eckart and Pöschl–Teller potentials.     

Darboux Transformations for Energy-Dependent Potentials and the Klein–Gordon Equation

We construct explicit Darboux transformations for a generalized Schrödinger-type equation with energy-dependent potential, a special case of which is the stationary Klein–Gordon equation. Our results complement and generalize former findings (Lin et al., Phys Lett A 362:212–214, 2007).     

Calculation of the Oscillator Strength for the Klein–Gordon Equation with Tietz Potential

The Klein–Gordon equation under equal scalar and vector potentials is solved for the Tietz potential in D-dimensions by using supersymmetric quantum mechanics. The spectrum of the system is numerically calculated and the oscillator strength is determined and discussed in terms of parameters of the system.     

Analytical Solution of the Klein–Fock–Gordon Equation for the Modified Klingberg Plus Ring Shaped Potentials

Russian Physics Journal - In this study, the bound state solution of the modified Klein–Fock–Gordon equation is found for new combined Klingberg and ring shaped potentials. Analytical...     

Exactly Solvable Combinations of Scalar and Vector Potentials for the Dirac Equation Interrelated by Riccati Equations

New classes of solvable scalar and vector potentials for the Dirac equation are obtained, together with the associated exact Dirac spinors. The method of derivation is based on an a priori constraint between the solutions, leading to an interrelation between the scalar and vector potential in the form ofa Riccati equation. The present note generalizes a series of former articles.     

Exact Solutions of the Two-Dimensional Schrödinger Equation with Certain Central Potentials

By applying an ansatz to the eigenfunction, an exact closed-form solution of theSchrödinger equation in two dimension is obtained with the potentials V(r) =ar ² + br ⁴ + cr ⁶,V(r) = ar + br² + cr ^–1,and V(r) = ar ² + br ^–2+ cr ^–4 + dr ^–6,respectively. The restrictions on the parameters of the given potential andthe angular momentum m are obtained.     

Exact Analytical Solution of the Klein–Gordon Equation in the Generalized Woods–Saxon Potential

The exact analytical solution of the Klein–Gordon equation for the spin-0 particles in the generalized Woods–Saxon potential is presented. The bound state energy eigenvalues and corresponding wave functions are obtained in the closed forms. The correlations between the potential parameters and energy eigenvalues are examined for π0particles.     

Analytical Solutions to the Klein-Gordon Equation with Position-Dependent Mass for q-Parameter P？schl-Teller Potential

The energy eigenvalues and the corresponding eigenfunctions of the one-dimensional Klein-Gordon equation with q-parameter P？schl-Teller potential are analytically obtained within the position-dependent mass formalism. The parametric generalization of the Nikiforov-Uvarov method is used in the calculations by choosing a mass distribution.     

Exact Polynomial Solutions of Schrdinger Equation with Various Hyperbolic Potentials

The Schrdinger equation with hyperbolic potential 2V(x) =-V0sinhq(x/d)/cosh6(x/d)(q = 0, 1, 2, 3) is studied by transforming it into the confluent Heun equation. We obtain general symmetric and antisymmetric polynomial solutions of the Schrdinger equation in a unified form via the Functional Bethe ansatz method. Furthermore, we discuss the characteristic of wavefunction of bound state with varying potential strengths. Particularly, the number of wavefunction's nodes decreases with the increase of potential strengths, and the particle tends to the bottom of the potential well correspondingly.     

Bound States for a Klein–Gordon Particle in Vector Plus Scalar Generalized Hulthén Potentials in D Dimensions

The Green’s function associated with a Klein–Gordon particle moving in a D-dimensional space under the action of vector plus scalar q-deformed Hulthén potentials is constructed by path integration for \({q \geq 1}\) and \({\frac{1}{\alpha} \ln q < r < \infty}\). An appropriate approximation of the centrifugal potential term and the technique of space-time transformation are used to reduce the path integral for the generalized Hulthén potentials into a path integral for q-deformed Rosen–Morse potential. Explicit path integration leads to the radial Green’s function for any l state in closed form. The energy spectrum and the correctly normalized wave functions, for a state of orbital quantum number \({l \geq 0}\), are obtained. Eventually, the vector q-deformed Hulthén potential and the Coulomb potentials in D dimensions are considered as special cases.     

Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes

For any sub-extremal Kerr spacetime with non-zero angular momentum, we find an open family of non-zero masses for which there exist smooth, finite energy, and exponentially growing solutions to the corresponding Klein–Gordon equation. If desired, for any non-zero integer m, an exponentially growing solution can be found with mass arbitrarily close to \({\frac{\left|am\right|}{2Mr_+}}\) . In addition to its direct relevance for the stability of Kerr as a solution to the Einstein–Klein–Gordon system, our result provides the first rigorous construction of a superradiant instability. Finally, we note that this linear instability for the Klein–Gordon equation contrasts strongly with recent work establishing linear stability for the wave equation.     

Analytical Solution of the Klein–Fock–Gordon Equation for the Rosen–Morse Potential

Russian Physics Journal - With the help of a successful scheme for overcoming difficulties arising for l ≠ 0 in the centrifugal part of the Rosen–Morse potential with bound states, a...     

Bounded Solutions of the Dirac Equation with a PT-symmetric Kink-Like Vector Potential in Two-Dimensional Space-Time

The (1+1)-dimensional Dirac equation with a PT-symmetric kink-like vector potential is investigated. By using the basic concepts of the supersymmetric WKB formalism and the function analysis method, we solve exactly the Dirac equation and obtain the bound-state energy levels and two-component spinor components. The PT-symmetric kink-like potential is not Hermitian and absent of bound states in the context of non-relativistic Schrödinger equation, but it possesses two sets of real discrete relativistic energy spectra in the context of the Dirac theory. When the PT symmetry is spontaneously broken, two sets of real energy spectra come into complex conjugate.     

Solution of the Dirac Equation with Position-Dependent Mass for q-Parameter Modified P?schl–Teller and Coulomb-like Tensor Potential

In this research, we have been obtained the Dirac equation for q-parameter modified P?schl–Teller potential including a Coulomb-like tensor interaction with arbitrary spin-orbit coupling quantum number by choosing a position-dependent mass. The energy eigenvalues equation and the corresponding unnormalized wave functions have been obtained. The Nikiforov-Uvarov method has been used in the calculations.     

Bound States of the S-Wave Equation with Equal Scalar and Vector Standard Eckart Potential

A supersymmetric technique for the bound-state solutions of the s-wave Klein-Gordon equation with equal scalar and vector standard Eckart-type potential is proposed. Its exact solutions are obtained. Possible generalization of our approach is outlined.     

Spectral Theory of the Klein–Gordon Equation in Pontryagin Spaces

In this paper we investigate an abstract Klein–Gordon equation by means of indefinite inner product methods. We show that, under certain assumptions on the potential which are more general than in previous works, the corresponding linear operator A is self-adjoint in the Pontryagin space induced by the so-called energy inner product. The operator A possesses a spectral function with critical points, the essential spectrum of A is real with a gap around 0, and the non-real spectrum consists of at most finitely many pairs of complex conjugate eigenvalues of finite algebraic multiplicity; the number of these pairs is related to the ‘size’ of the potential. Moreover, A generates a group of bounded unitary operators in the Pontryagin space . Finally, the conditions on the potential required in the paper are illustrated for the Klein–Gordon equation in ; they include potentials consisting of a Coulomb part and an L _p -part with n ≤ p < ∞.Branko Najman: Deceased