  We consider second order differential equations with real coefficients that
are in the limit circle case at infinity. Using the semiclassical Ansatz, we
construct solutions (the Jost solutions) of such equations with a prescribed
asymptotic behavior for $x\to\infty$. It turns out that in the limit circle
case, this Ansatz can be chosen common for all values of the spectral parameter
$z$. This leads to asymptotic formulas for all solutions of considered
differential equations, both homogeneous and non-homogeneous. We also
efficiently describe all self-adjoint realizations of the corresponding
differential operators in terms of boundary conditions at infinity and
  find a representation for their resolvents.
