  This work investigates spectrum and root functions (that is, eigen- and
associated functions) of a Sturm-Liouville problem involving an abstract linear
operator (nonselfadjoint in general) in the equation together with
supplementary transmission conditions at the one interior singular point. So,
the problem under consideration is not pure differential pro lem. At first we
establish isomorphism and coerciveness with respect to the spectral parameter
for the corresponding nonhomogeneous problem. Then by suggesting an own our
method we prove that the spectrum of the considered problem is discrete and
derive an asymptotic approximation formulas for the eigenvalues. We must note
that Asymptotics of the eigenvalues of such type problems is investigated at
first in literature in the present work and the obtained results are new even
in the continuous case (i.e. without transmission conditions). Finally it is
shown that the system of root functions form an Abel basis of the corresponding
Hilbert space.
