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The Bethe-Salpeter Equation for a bound system, composed by two massivescalars exchanging a massive scalar, is quantitatively investigated in ladderapproximation, within the Nakanishi integral representation approach. For theS-wave case, numerical solutions with a form inspired by the Nakanishi integralrepresentation, have been calculated. The needed Nakanishi weight functionshave been evaluated by solving two different eigenequations, obtained directlyfrom the Bethe-Salpeter equation applying the Light-Front projection technique.A remarkable agreement, in particular for the eigenvalues, has been achieved,numerically confirming that the Nakanishi uniqueness theorem for the weightfunctions, demonstrated in the context of the perturbative analysis of themulti-leg transition amplitudes and playing a basic role in suggesting one ofthe two adopted eigenequations, can be extended to a non perturbative realm.The detailed, quantitative studies are completed by presenting bothprobabilities and Light-Front momentum distributions for the valence componentof the bound state.

Quantitative studies of the homogeneous Bethe-Salpeter Equation in Minkowski space