  We develop a general approach to calculations of the electron spectrum of
metallic multi-wall carbon nanotubes (MWNT) with arbitrary number of coaxial
layers. It is based on the model with singular attractive potential of
equidistant conductive cylinders. The knowledge of one-electron spectrum allows
to construct the corresponding Green function and then to calculate the entropy
and density of states for MWNT. We analyze the tunnelling between the nanotube
and normal metal electrode. The possibility of direct determination of
one-electron density of states by measurements of the tunnelling conductivity
at low temperatures is proved and the necessary restrictions on temperature are
formulated. We discuss briefly the conflicting experimental observations of
electronic properties of MWNT.
