  In frames of the nonlocal and nonpolynomial quantum theory of the one
component scalar field in $D$-dimensional spacetime, stated by Gariy
Vladimirovich Efimov, the expansion of the $\mathcal{S}$-matrix is revisited
for different interaction Lagrangians and for some kinds of Gaussian
propagators modified by different ultraviolet form factors $F$ which depend on
some length parameter $l$. The expansion of the $\mathcal{S}$-matrix is of the
form of a grand canonical partition function of some $D+N$-dimensional ($N\geq
1$) classical gas with interaction. The toy model of the realistic quantum
field theory (QFT) is considered where the $\mathcal{S}$-matrix is calculated
in closed form. Then, the functional Schwinger-Dyson and Schr\"{o}dinger
equations for the $\mathcal{S}$-matrix in Efimov representation are derived.
These equations play a central role in the present paper. The functional
Schwinger-Dyson and Schr\"{o}dinger equations in Efimov representation do not
involve explicit functional derivatives but involve a shift of the field which
is the $\mathcal{S}$-matrix argument. The asymptotic solutions of the
Schwinger-Dyson equation are obtained in different limits. Also, the solution
is found in one heuristic case allowing us to study qualitatively the behavior
of the $\mathcal{S}$-matrix for an arbitrary finite value of its argument.
Self-consistency equations, which arise during the process of derivation, are
of a great interest. Finally, in the light of the discussion of QFT functional
equations, ultraviolet form factors and extra dimensions, the connection with
functional (in terms of the Wilson-Polchinski and Wetterich-Morris functional
equations) and holographic renormalization groups (in terms of the functional
Hamilton-Jacobi equation) is made. In addition the Hamilton-Jacobi equation is
formulated in an unconventional way.
