Separation of variables for the Klein–Gordon and Dirac equations in two-dimensional space-times with a null coordinate

1990 ◽  
Vol 68 (11) ◽  
pp. 1243-1246
Author(s):  
Victor M. Villalba
Keyword(s):  
Dimensional Space ◽  
Separation Of Variables ◽  
Constant Electric Field ◽  
Coordinate Systems ◽  
Constant Acceleration ◽  
Quantum Distribution ◽  
Dirac Equations ◽  
Curvilinear Coordinate ◽  
Klein Gordon ◽  
Free Case

The separation of variables for the Klein–Gordon and Dirac equations, in the presence of electromagnetic fields, for a class of curvilinear coordinate systems with a null coordinate is presented. We show that these coordinates can be associated with a system with constant acceleration. Exact solutions for the free case and for a particle in a constant electric field are obtained. Finally, the quantum distribution of scalar massless particles in the accelerated frame of reference is computed.

SEPARATING COORDINATE SYSTEMS IN THE MINKOWSKIAN SPACE-TIME

1991 ◽  
Vol 06 (17) ◽  
pp. 3109-3117
Author(s):  
Z.Y. TURAKULOV
Keyword(s):  
Separation Of Variables ◽  
Complete Separation ◽  
Space Time ◽  
Coordinate Systems ◽  
Field Equations ◽  
Minkowskian Space ◽  
Curvilinear Coordinate ◽  
Klein Gordon ◽  
Scalar Field Equations ◽  
Variable Separation Method

A class of metrics providing the complete separation of variables in the Klein-Gordon equation is considered. The general exact solution to the vacuum Einstein equation for such metrics is obtained by the variable separation method. It is shown that all these solutions correspond to curvilinear coordinate systems in the Minkowskian space-time. Several limit cases of such systems are investigated. Moreover, some other separating systems are constructed and it is shown that they make it possible to obtain partial exact solutions to nonlinear scalar field equations.

Vacuum effects in a spatially homogeneous and isotropic cosmological background

1992 ◽  
Vol 70 (2-3) ◽  
pp. 143-147
Author(s):  
Victor M. Villalba ◽  
Umberto Percoco
Keyword(s):  
Separation Of Variables ◽  
Particle Creation ◽  
Coupling Constant ◽  
Positive Cosmological Constant ◽  
Dirac Equations ◽  
Klein Gordon ◽  
Dimensionless Coupling ◽  
Spatially Homogeneous ◽  
Dimensionless Coupling Constant ◽  
Vacuum Effects

In this article we obtain, by separation of variables, an exact solution to the Klein–Gordon and Dirac equations in a cosmological, spatially-closed, Robertson–Walker space-time with a positive cosmological constant Λ. The model is associated with a universe filled with radiation. We analyze the phenomenon of particle creation for different values of the dimensionless coupling constant ξ.

Exact solution of Klein–Gordon equation in fractional-dimensional space

2021 ◽  
Author(s):  
H. Merad ◽  
F. Merghadi ◽  
A. Merad
Keyword(s):  
Exact Solution ◽  
Gordon Equation ◽  
Special Functions ◽  
Dimensional Space ◽  
Confluent Hypergeometric Functions ◽  
Heisenberg Algebras ◽  
Klein Gordon ◽  
Free Case ◽  
Coulomb Type ◽  
Klein Gordon Equation

In this paper, we present an exact solution of the Klein–Gordon equation in the framework of the fractional-dimensional space, in which the momentum and position operators satisfying the R-deformed Heisenberg algebras. Accordingly, three essential problems have been solved such as: the free Klein–Gordon equation, the Klein–Gordon equation with mixed scalar and vector linear potentials and with mixed scalar and vector inversely linear potentials of Coulomb-type. For all these considered cases, the expressions of the eigenfunctions are determined and expressed in terms of the special functions: the Bessel functions of the first kind for the free case, the biconfluent Heun functions for the second case and the confluent hypergeometric functions for the end case, and the corresponding eigenvalues are exactly obtained.

scholarly journals QUANTUM SINGULARITIES IN STATIC SPACETIMES

2011 ◽  
Vol 20 (05) ◽  
pp. 729-743 ◽  
Author(s):  
JOÃO PAULO M. PITELLI ◽  
PATRICIO S. LETELIER
Keyword(s):  
Quantum Mechanics ◽  
Wave Operator ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Mathematical Framework ◽  
Physical Content ◽  
Dirac Equations ◽  
Quantum Particles ◽  
Klein Gordon ◽  
Quantum Singularities

We review the mathematical framework necessary to understand the physical content of quantum singularities in static spacetimes. We present many examples of classical singular spacetimes and study their singularities by using wave packets satisfying Klein–Gordon and Dirac equations. We show that in many cases the classical singularities are excluded when tested by quantum particles but unfortunately there are other cases where the singularities remain from the quantum mechanical point of view. When it is possible we also find, for spacetimes where quantum mechanics does not exclude the singularities, the boundary conditions necessary to turn the spatial portion of the wave operator to be self-adjoint and emphasize their importance to the interpretation of quantum singularities.

scholarly journals Implementation of an Approximate Conformal UPML in 2-D DGTD

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Linqian Li ◽  
Bing Wei ◽  
Qian Yang ◽  
Debiao Ge
Keyword(s):  
Good Accuracy ◽  
Computation Time ◽  
Elliptic Cylinder ◽  
Auxiliary Equation ◽  
Coordinate Systems ◽  
Absorbing Boundary ◽  
Computational Region ◽  
Curvilinear Coordinate ◽  
Orthogonal Curvilinear Coordinate System ◽  
Absorbing Layer

Using the numerical discrete technique with unstructured grids, conformal perfectly matched layer (PML) absorbing boundary in the discontinuous Galerkin time-domain (DGTD) can be set flexibly so as to save lots of computing resources. Based on the DGTD equations in an orthogonal curvilinear coordinate system, the processes of parameter transformation for 2-D UPML between the coordinate systems of elliptical and Cartesian are given; and the expressions of transition matrix are derived. The calculation scheme of conductivity distribution in elliptic cylinder absorbing layer is given, and the calculation coefficient of DGTD in elliptic UPML is calculated. Furthermore, the 2-D iterative formulas of DGTD and that of auxiliary equation in the elliptical cylinder UPML are derived; the conformal UPML calculation in DGTD is realized. Numerical results show that very good accuracy and computational efficiency are achieved by using the method in this paper. Compared to the rectangular computational region, both the memory and computation time of conformal UPML absorbing boundary are reduced by more than 20%.

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