  The article suggests a method for the construction of the number of atoms
preserving microscopic He II theory. The suggested theory can provide the
ground state wave function (WF) as an expansion in series by small parameters.
In addition, errors in the definition of the WFs of the excited states turn out
to be vanishingly small while the system size increases. Predictions of the
proposed theory and of the Bogoliubov theory are identical if helium occupies a
simply connected volume. However, there are differences in the general case.
These differences are due to the fact that the operator of the occupation
number of the zero momentum state does not commute with the creation and
annihilation operators of phonons. This contradicts the Bogoliubov assumption
that the creation and annihilation operators of atoms in the zero momentum
state can be replaced by a c-number. The latter, as is known, should commute
with any operator. This should lead to a difference in the results of
calculation of helium flow through the tunnel transition. Therefore, the
suggested theory is not equivalent to Bogoliubov theory.

Superconductivity;Quantum Gases