We apply the perturbation theory integral representation (PTIR) to solve for
the bound state Bethe-Salpeter (BS) vertex for an arbitrary scattering kernel,
without the need for any Wick rotation. The results derived are applicable to
any scalar field theory (without derivative coupling). It is shown that solving
directly for the BS vertex, rather than the BS amplitude, has several major
advantages, notably its relative simplicity and superior numerical accuracy. In
order to illustrate the generality of the approach we obtain numerical
solutions using this formalism for a number of scattering kernels, including
cases where the Wick rotation is not possible.Comment: 28 pages of LaTeX, uses psfig.sty with 5 figures. Also available via
WWW at
http://www.physics.adelaide.edu.au/theory/papers/ADP-97-10.T248-abs.html or
via anonymous ftp at
ftp://bragg.physics.adelaide.edu.au/pub/theory/ADP-97-10.T248.ps A number of
(crucial) typographical errors in Appendix C corrected. To be published in
Phys. Rev. D, October 199
