Salpeter equation in position space: Numerical solution for arbitrary confining potentials
1984; American Physical Society; Volume: 30; Issue: 3 Linguagem: Inglês
10.1103/physrevd.30.660
ISSN1538-4500
AutoresL. J. Nickisch, Loyal Durand, Bernice Durand,
Tópico(s)Black Holes and Theoretical Physics
ResumoWe present and test two new methods for the numerical solution of the relativistic wave equation $[{(\ensuremath{-}{\ensuremath{\nabla}}^{2}+{{m}_{1}}^{2})}^{\frac{1}{2}}+{(\ensuremath{-}{\ensuremath{\nabla}}^{2}+{{m}_{2}}^{2})}^{\frac{1}{2}}+V(r)\ensuremath{-}M]\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=0$, which appears in the theory of relativistic quark-antiquark bound states. Our methods work directly in position space, and hence have the desirable features that we can vary the potential $V(r)$ locality in fitting the $q\overline{q}$ mass spectrum, and can easily build in the expected behavior of $V$ for $r\ensuremath{\rightarrow}0$,$\ensuremath{\infty}$. Our first method converts the nonlocal square-root operators to mildly singular integral operators involving hyperbolic Bessel functions. The resulting integral equation can be solved numerically by matrix techniques. Our second method approximates the square-root operators directly by finite matrices. Both methods converge rapidly with increasing matrix size (the square-root matrix method more rapidly) and can be used in fast-fitting routines. We present some tests for oscillator and Coulomb interactions, and for the realistic Coulomb-plus-linear potential used in $q\overline{q}$ phenomenology.
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