Theory of a degenerate Fermi gas in an external field

The variation in the density $\delta\rho(x,E)$ of a degenerate ideal Fermi gas is found for the case of a localized variation of an external field $V(x)$. It is shown that the continuous dependence of $\delta\rho$ on $\delta V$ is not violated if discrete levels far from the Fermi level $E$ arise (or disappear). In particular, if there is ,an energy gap $(E_1, E_2)$ and $E_1<E<E_2$, the occurrence of discrete levels does not reduce the rate of exponential decrease of $\delta\rho(x,E)$ as $|x|\to\infty$.