A solvable two-body Dirac equation in one space dimension

1991 ◽  
Vol 69 (7) ◽  
pp. 780-785 ◽  
Author(s):  
F. Dominguez-Adame ◽  
B. Méndez
Keyword(s):  
Wave Function ◽  
Dirac Equation ◽  
Space Dimension ◽  
Translational Motion ◽  
Effective Frequency ◽  
Klein Paradox ◽  
Independent Components ◽  
Dirac Particles ◽  
One Space Dimension

A solvable Hamiltonian for two Dirac particles interacting by instantaneous linear potentials in (1 + 1) dimensions is discussed. The system presents no Klein paradox even if the coupling is rather strong, so particles remain bound. The four independent components of the wave function describing the system resemble the nonrelativistic oscillator eigenfunctions. Although the Hamiltonian is not fully covariant, the effective frequency of the oscillator obeys a typical relativistic Doppler law. In contrast to the nonrelativistic treatment, eigenstates are intrinsically coupled with the overall translational motion of the system.

Two-body Dirac equation: illustration in one space dimension

1988 ◽  
Vol 66 (9) ◽  
pp. 769-775 ◽  
Author(s):  
F. A. B. Coutinho ◽  
W. Glöckle ◽  
Y. Nogami ◽  
F. M. Toyama
Keyword(s):  
Dirac Equation ◽  
Space Dimension ◽  
Bound State ◽  
Sharp Edge ◽  
Finite Range ◽  
Range Interaction ◽  
Square Well ◽  
The One ◽  
The Masses ◽  
One Space Dimension

Various features that are characteristic of the two-body Dirac equation but not of the one-body Dirac equation are illustrated by means of solvable examples (mainly square-well potentials) in one space dimension. For the Lorentz character of the potential there are three types; vector, scalar, and pseudoscalar. We classify the bound-state solutions as normal and abnormal. As the interaction is adiabatically switched off, the energy of the normal solutions reaches 2m (the sum of the masses of the constituent particles), whereas the energy of the abnormal solutions becomes zero. When the sharp edge of the square-well potential is smeared out, some of the solutions become unnormalizable and hence unacceptable. This leads to a certain restriction on the choice of the potential. The two-body Dirac equation with a finite-range interaction is not exactly covariant. The degree of noncovariance is examined.

DIRAC EQUATION WITH CERTAIN QUADRATIC NONLINEARITIES IN ONE SPACE DIMENSION

2007 ◽  
Vol 09 (03) ◽  
pp. 421-435 ◽  
Author(s):  
SHUJI MACHIHARA
Keyword(s):  
Dirac Equation ◽  
Space Dimension ◽  
Fourier Transforms ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Existence Results ◽  
Bilinear Estimates ◽  
Quadratic Nonlinearities ◽  
Local Existence Of Solutions ◽  
One Space Dimension

We discuss the time local existence of solutions to the Dirac equation for special types of quadratic nonlinearities in one space dimension. Solutions with more rough data than those of the previous work [15] are obtained. The Fourier transforms of solutions with respect to both variables x and t are investigated. Certain linear and bilinear estimates on solutions are derived, and a standard iteration argument gives the existence results.

scholarly journals Self-adjoint Dirac type Hamiltonians in one space dimension with a mass jump

2015 ◽  
Vol 48 (4) ◽  
pp. 045207 ◽  
Author(s):  
L A González-Díaz ◽  
Alberto A Díaz ◽  
S Díaz-Solórzano ◽  
J R Darias
Keyword(s):  
Space Dimension ◽  
Dirac Type ◽  
One Space Dimension

scholarly journals LOW REGULARITY WELL-POSEDNESS OF THE DIRAC–KLEIN–GORDON EQUATIONS IN ONE SPACE DIMENSION

2008 ◽  
Vol 10 (02) ◽  
pp. 181-194 ◽  
Author(s):  
SIGMUND SELBERG ◽  
ACHENEF TESFAHUN
Keyword(s):  
Dirac Operator ◽  
Space Dimension ◽  
One Dimension ◽  
Well Posedness ◽  
Low Regularity ◽  
Klein Gordon ◽  
One Space Dimension

We extend recent results of Machihara and Pecher on low regularity well-posedness of the Dirac–Klein–Gordon (DKG) system in one dimension. Our proof, like that of Pecher, relies on the null structure of DKG, recently completed by D'Ancona, Foschi and Selberg, but we show that in 1d the argument can be simplified by modifying the choice of projections for the Dirac operator. We also show that the result is best possible up to endpoint cases, if one iterates in Bourgain–Klainerman–Machedon spaces.

Sign in / Sign up

Export Citation Format

Share Document