| |
| This paper solves Klein-Gordon equation (KGE) with the direct coupling of linear vector and scalar potentials, viz $V(r)=V_0r$ and $S(r)=S_0r$ with the order $V_0<S_0$ on $mathbb{R}+$. The work begins with the general formulation of KGE under the coupling of a vector and a scalar potential. Firstly, the $s$-wave KGE in coupling linear potentials is solved. Transformations lead the dynamical equation into parabolic cylinder equation, Webers equation and confluent hypergeometric equation, respectively, and these three approaches give rise to the identical energy spectrum and eigenfunction. Subsequently, the coupling KGE with nonvanishing centrifugal effect is calculated via Frobenius method of series expansion, which firstly yields the recurrence relation of the series coefficients.
Since the indicial indexes are $s_1=-l$, $s_2=l+1$, when $l=0$, the solution belongs to the special case with indeterminate coefficient; when $l geq1$ with centrifugal effect, one encounters the special case with integer index difference. These two cases share the same $z(r,s)$ kernel of solution, and in the $l=0$ case, emphasis is shifted to the resolved $s$-wave states, while in the $l geq 1$ case, the kernel $z(r,s)$ and the asymptotic eigenfunction are established. Its argued that the theory of exactly-solvable high-order difference equation should be further developed.
[(Comment: 12 pages, no figure)] |
|
| Keywords:Klein-Gordon Equation;Linear Potential;Bound State;Special Function;Frobenius Method;Difference Equation |
|
