Integral equation for the causal distributions and their self-similar asymptotic behavior in the ladder $\Phi^3$ model

A method is proposed for deriving an integral equation for the spectral density of the causal distributions of the single-particle diagonal matrix element of the commutator of scalar currents. In the ladder approximation of the $\Phi^3$ model in the case of a massless exchange particle, the equation is solved exactly. Closed expressions are found for the spectral functions of the Deser–Gilbert–Sudarshan and Jost–Lehmann–Dyson representations for the single-particle matrix element of the current commutator.