  Analytical solutions of the edge states were obtained for the (N, 0) type
carbon nanotubes with distorted ending bonds. It was found that the edge states
are mixed via the distortion. The total energies for N=5 and N>=7 are lower in
the asymmetric configurations of ending bonds than those having axial rotation
symmetry. Thereby the symmetry is breaking spontaneously. The results imply
that the symmetry of electronic states at the apex depends on the occupation;
the electron density pattern at the apex could change dramatically and could be
controlled by applying an external field.
