Exact solution of Bethe–Salpeter equation with retarding propagators in Wick–Cutkosky model

We investigate the discrete spectrum of the Bethe–Salpeter equation for the Wick–Cutkosky model. It is shown that in the case of retarding propagators the equation becomes separable, and in the $S$-wave case it is reduced to a one-dimensional equation. Using the method of countour integration, we obtain the exact solution of this equation and investigate the spectrum of relativistic bound states. We discuss the ambiguity of analytical expressions for the wave function, which originates from the occurence of unphysical variable in the problem of four-dimensional description of relativistic systems.