  We present exact solution of the problem of electronic wave functions of
quasi one-dimensional band with an inter-band gap at the Fermi surface and in
the presence of magnetic field. The details of the analyzed model are
appropriate to the situation in the Bechgaard salt (TMTSF)2ClO4 with the
dimerizing anion order in the transverse direction. Limiting the effects of
dimerization to the standard dimerization gap only, one obtains the electronic
spectrum represented through solutions of a generalized Hill system of
equations with simply periodic coefficients. The resulting wave-functions are
discussed. In particular, we present the solutions for the case when the
electrons spend as much time in the "junctions" as on their quasi-classical
orbits. On the other hand, the limit when the tunnelling approach is valid is
identified and the results are confronted with the well-known
Slutskin-Kadigrobov solution. Furthermore, taking into account also the
presumably finite transverse dimerizing displacements of chains, one encounters
the qualitatively more complex problem of a system of equations with
two-periodic coefficients. Some qualitatively new properties of electronic
spectrum and corresponding one-electron physical quantities in this case will
be discussed in detail.
