  We investigate theoretically the eigenmodes and the stability of one and two
arbitrary fractional vortices pinned at one and two $\kappa$-phase
discontinuities in a long Josephson junction. In the particular case of a
single $\kappa$-discontinuity, a vortex is spontaneously created and pinned at
the boundary between the 0 and $\kappa$-regions. In this work we show that only
two of four possible vortices are stable. A single vortex has an oscillatory
eigenmode with a frequency within the plasma gap. We calculate this
eigenfrequency as a function of the fractional flux carried by a vortex.
  For the case of two vortices, pinned at two $\kappa$-discontinuities situated
at some distance $a$ from each other, splitting of the eigenfrequencies occur.
We calculate this splitting numerically as a function of $a$ for different
possible ground states. We also discuss the presence of a critical distance
below which two antiferromagnetically ordered vortices form a strongly coupled
``vortex molecule'' that behaves as a single object and has only one eigenmode.
