Application of some algebraic arguments to the theory of normal Fermi systems

A study is made of the many-particle operator
$$ H=\sum_1^N(-\Delta_k)+\sum_{i<j}V(|x_i-x_j|). $$
Under the assumption that the system of Fermi particles in a volume $\Omega$, $N/\Omega=p$, it is shown that there exists a formal operator series $S$ such that $HS\Psi_{\alpha}=SF\Psi_{\alpha}$, where $F$ is a function of only the occupation number operators, and $\Psi_{\alpha}=a^*_{\alpha_1}\dots a^*_{\alpha_N}\Psi^0$, where ,$\Psi^0$ – is the vacuum vector. The series $F$ is a series in powers of the density; knowledge of $F$ makes it possible to calculate the Gibbs potential at “low” densities but in the whole temperature range. The relation to the Landau theory of normal Fermi fluids is discussed. The first two terms of the series for $F$ are calculated. A number of questions relating to the passage to the limit $\Omega\to\infty$ is discussed.