  We develop a continuous theory of low-frequency dynamics for single-walled
carbon nanotubes (SWCNTs) weakly interacting with the environment. In the frame
of the approach proposed we obtain temperature dependence of SWCNTs specific
heat in the low (T<40 K) and ultra-low (T<2.5 K) temperature ranges. We take
into account the main term in the coupling between SWCNT and the environment
that slightly increases the frequencies of those SWCNT modes, which possess
predominantly radial polarization. The coupling drastically decreases the
density of phonon states in the lowest frequencies region. The theoretically
predicted fall of the specific heat in the interval T<2.5 K properly explains
available experimental data in contrast to the preceding approaches. The theory
proposed can be the basis for studies of low-temperature heat capacity and
phonon dynamics of many other single-walled and multi-walled tubular structures
(boron nitride, transition metal dichalcogenides) which have emerged in the
past decade.
