  The approach applicable for spatially inhomogeneous and time-dependent
problems associated with the induced superconductivity in low dimensional
electronic systems is developed. This approach is based on the Fano--Anderson
model which describes the decay of a resonance state coupled to a continuum. We
consider two types of junctions made of a ballistic 2D electron gas placed in a
tunnel finite-length contact with a bulk superconducting leads. We calculate
the spectrum of the bound states, supercurrent, and the current-voltage curve
which show a rich structure due to the presence of induced gap and dimensional
quantization.
