Microscopic theory of the $\lambda$-transition in liquid $\mathrm{He}^4$. I

An exact expression derived earlier by the author for the statistical operator of a manyboson system in the representation of coherent states is used to give a microscopic justification of the effective functional of the thermodynamic potential describing the fluctuations of the order parameter in the theory of the $\lambda$ transition of liquid $\mathrm{He}^4$. Using the method of functional integration under the assumption that the critical exponent is zero, $\eta=0$, a recursion relation is obtained for the functional of the thermodynamic potential in the case of an $n$-component order parameter. It is shown for the example of the $\psi^4$ model that the recursion formula gives correct perturbation series. The connection between the recursion formula obtained here and Wilson's well-known approximate recursion relation is discussed.