  This paper is the first one of a series of two articles in which we revisit
the optical properties of single-walled carbon nanotubes (SWNT). Produced by
rolling up a graphene sheet, SWNT owe their intriguing properties to their
cylindrical quasi-one-dimensional (quasi-1D) structure (the ratio length/radius
is experimentally of order of 10^3). We model SWNT by circular cylinders of
small diameters on the surface of which the conduction electron gas is confined
by the electric field generated by the fixed carbon ions. The pair-interaction
potential considered is the 3D Coulomb potential restricted to the cylinder. To
reflect the quasi-1D structure, we introduce a 1D effective many-body
Hamiltonian which is the starting-point of our analysis. To investigate the
optical properties, we consider a perturbation by a uniform time-dependent
electric field modeling an incident light beam along the longitudinal
direction. By using Kubo's method, we derive within the linear response theory
an asymptotic expansion in the low-temperature regime for the dynamical optical
conductivity at fixed density of particles. The leading term only involves the
eigenvalues and associated eigenfunctions of the (unperturbed) 1D effective
many-body Hamiltonian, and allows us to account for the sharp peaks observed in
the optical absorption spectrum of SWNT.
