  We study the singularities in X-ray absorption spectra of one-dimensional
Hubbard and $t$--$J$ models. We use Boundary Conformal Field Theory and the
Bethe Ansatz solutions of these models with both periodic and open boundary
conditions to calculate the exponents describing the power-law decay near the
edges of X-ray absorption spectra in the case where the core-hole potential has
bound states.
