Investigation of zero-frequency solutions to the pion dispersion equation

The instability of unclear matter is considered for the case where it is generated by the vanishing of the frequencies of collective excitations belonging to specific types (specifically, excitations that have the pion quantum numbers J(pi) = 0(-)). The behavior of zero-frequency solutions to the pion dispersion equation is analyzed versus the strength G' of spin-isospin particle-hole interaction. It is shown that there exists a strength value G'(tr) (vertical bar G'tr vertical bar << 1) Such that, for G' < G'(tr), zero-frequency solutions are excitations of the omega(p) type, while, for G' >= G'(tr), such solutions are excitations of the omega(c) type. Excitations of the omega(p) type for G' < -1 describe the instability of nuclear matter against small density fluctuations (Pomeranchuk's instability), while excitations of the omega(c) type are responsible for the instability associated with pion condensation at G' approximate to 2. For stable unclear matter, the solutions omega(1),(k) and omega(c),(k) lie on unphysical sheets of the complex plane of frequency.