  We consider a fractional generalization of two-dimensional (2D)
quantum-mechanical Kepler problem corresponding to 2D hydrogen atom. Our main
finding is that the solution for discreet spectrum exists only for $\mu>1$
(more specifically $1 < \mu \leq 2$, where $\mu=2$ corresponds to "ordinary" 2D
hydrogenic problem), where $\mu$ is the L\'evy index. We show also that in
fractional 2D hydrogen atom, the orbital momentum degeneracy is lifted so that
its energy starts to depend not only on principal quantum number $n$ but also
on orbital $m$. To solve the spectral problem, we pass to the momentum
representation, where we apply the variational method. This permits to obtain
approximate analytical expressions for eigenvalues and eigenfunctions with very
good accuracy. Latter fact has been checked by numerical solution of the
problem. We also found the new integral representation (in terms of complete
elliptic integrals) of Schr\"odinger equation for fractional hydrogen atom in
momentum space. We point to the realistic physical systems like bulk
semiconductors as well as their heterostructures, where obtained results can be
used.
