Autocorrelation functions of an isotropic paramagnetic Fermi liquid at Landau attenuation threshold

The calculation of the autocorrelation functions of density (spin density) and transverse current (spin current) for a degenerate Fermi liquid is reduced to the algebraic problem of inversion of a particular matrix. This approach is used to obtain the autocorrelation function asymptotics at the Landau attenuation threshold, i.e., for ω/kv ≡ μ → l, where ω and k are the frequency and wave vector of the external field, and v is the Fermi velocity. It is proved that the autocorrelation functions of density (spin density) and transverse current (spin current) are linear fractional functions of the variables q = ln[(l − μ/2 + i0] and (1 − μ)q under the assumption that terms of the order of (1 − μ)q and (1 − μ) are neglected in comparison to q and (1 − μ)q, respectively. The particular shape of the asymptotics is calculated for three nonzero Landau parameters. The formulas obtained may be used for studying the nonexponential decay (from the sample boundary) of fields in 3He and alkali metals.