  We study a set of crossed 1D systems, which are coupled with each other via
tunnelling at the crossings. We begin with the simplest case with no
electron-electron interactions and find that besides the expected level
splitting, bound states can emerge. Next, we include an external potential and
electron-electron interactions, which are treated within the Hartree
approximation. Then, we write down a formal general solution to the problem,
giving additional details for the case of a symmetric external potential.
Concentrating on the case of a single crossing, we were able to explain recent
experinents on crossed metallic and semiconducting nanotubes [J. W. Janssen, S.
G. Lemay, L. P. Kouwenhoven, and C. Dekker, Phys. Rev. B 65, 115423 (2002)],
which showed the presence of localized states in the region of crossing.
