We have studied the quantum motion of charge carriers through a period
ical system of atoms placed along a cylindrical spiral (of fullerene t
ubule kind) in the presence of an external magnetic field. Expressions
for a quasi-one-dimensional dispersion law have been obtained within
the framework of the modified tight-binding model. These expressions e
xhibit side minima and a nontrivial dependence on the magnetic field.
Possible experimental manifestations are discussed.