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Register a new userProper treatment of scalar and vector exponential potentials in the Klein-Gordon equation: Scattering and bound states
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We point out a misleading treatment in the literature regarding to bound-state solutions for the $s$-wave Klein-Gordon equation with exponential scalar and vector potentials. Following the appropriate procedure for an arbitrary mixing of scalar and vector couplings, we generalize earlier works and present the correct solution to bound states and additionally we address the issue of scattering states. Moreover, we present a new effect related to the polarization of the charge density in the presence of weak short-range exponential scalar and vector potentials.
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We solve the Klein-Gordon equation in any $D$-dimension for the scalar and vector general Hulth{e}n-type potentials with any $l$ by using an approximation scheme for the centrifugal potential. Nikiforov-Uvarov method is used in the calculations. We obtain the bound state energy eigenvalues and the corresponding eigenfunctions of spin-zero particles in terms of Jacobi polynomials. The eigenfunctions are physical and the energy eigenvalues are in good agreement with those results obtained by other methods for D=1 and 3 dimensions. Our results are valid for $q=1$ value when $l eq 0$ and for any $q$ value when $l=0$ and D=1 or 3. The $s$% -wave ($l=0$) binding energies for a particle of rest mass $m_{0}=1$ are calculated for the three lower-lying states $(n=0,1,2)$ using pure vector and pure scalar potentials.
In this work we study the Dirac equation with vector and scalar potentials in the spacetime generated by a cosmic string. Using an approximation for the centrifugal term, a solution for the radial differential equation is obtained. We consider the scattering states under the Hulth{e}n potential and obtain the phase shifts. From the poles of the scattering $S$-matrix the states energies are determined as well.
We consider the quantum mechanical problem of the motion of a spinless charged relativistic particle with mass$M$, described by the Klein-Fock-Gordon equation with equal scalar $S(vec{r})$ and vector $V(vec{r})$ Coulomb plus ring-shaped potentials. It is shown that the system under consideration has both a discrete at $left|Eright|<Mc^{2} $ and a continuous at $left|Eright|>Mc^{2} $ energy spectra. We find the analytical expressions for the corresponding complete wave functions. A dynamical symmetry group $SU(1,1)$ for the radial wave equation of motion is constructed. The algebra of generators of this group makes it possible to find energy spectra in a purely algebraic way. It is also shown that relativistic expressions for wave functions, energy spectra and group generators in the limit $cto infty $ go over into the corresponding expressions for the nonrelativistic problem.
The relativistic quantum motion of scalar bosons under the influence of a full vector (minimal $A^{mu}$ and nonminimal $X^{mu}$) and scalar ($V_{s}$) interactions embedded in the background of a cosmic string is explored in the context of the Klein-Gordon equation. Considering Coulomb interactions, the effects of this topological defect in equation of motion, phase shift and S-matrix are analyzed and discussed. Bound-state solutions are obtained from poles of the S-matrix and it is shown that bound-state solutions are possible only for a restrict range of coupling constants.
We consider the nonlinear Klein-Gordon equation in $R^d$. We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates.
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