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In order to solve the Bethe-Salpeter equation (BSE) in the Minkowski space,we first introduce the Nakanishi integral representations of the Bethe-Salpeteramplitude (BSA) and the Bethe-Salpeter wave function (BSWF). We then derive theexplicit integral equations for the corresponding spectral functions from theBSE for states of $2$ scalar particles bound by a scalar-particle exchangeinteraction, where the propagators of constituents are allowed to be fullydressed. These integral equations are subsequently solved numerically in thevariation of the Wick-Cutkosky model with massive exchange particles, where analgorithm of adaptive mash grid is proposed. The equations and algorithm wedevelop here serve as the foundation of Minkowski-space formulation of BSE forbound states of fermions.

Direct solution of Minkowski-space Bethe-Salpeter equation in the massive Wick-Cutkosky model