  The Bethe-Salpeter (BS) equation for scalar-scalar bound states in scalar
theories without derivative coupling is formulated and solved in Minkowski
space. This is achieved using the perturbation theory integral representation
(PTIR), which allows these amplitudes to be expressed as integrals over weight
functions and known singularity structures and hence allows us to convert the
BS equation into an integral equation involving weight functions. We obtain
numerical solutions using this formalism for a number of scattering kernels to
illustrate the generality of the approach. It applies even when the na\"{\i}ve
Wick rotation is invalid. As a check we verify, for example, that this method
applied to the special case of the massive ladder exchange kernel reproduces
the same results as are obtained by Wick rotation.
