  We analyze the impact of trap states in the oxide layer of a superconducting
tunnel junctions, on the fluctuation of the Josephson critical current, thus on
coherence in superconducting qubits. Two mechanisms are usually considered: the
current blockage due to repulsion at the occupied trap states, and the noise
from electrons hopping across a trap. We extend previous studies of
noninteracting traps to the case where the traps have on-site electron
repulsion inside one ballistic channel. The repulsion not only allows the
appropriate temperature dependence of 1/f noise, but also is a control to the
coupling between the computational qubit and the spurious two-level systems
inside the oxide dielectric. We use second order perturbation theory which
allows to obtain analytical formulae for the interacting bound states and
spectral weights, limited to small and intermediate repulsions. Remarkably, it
still reproduces the main features of the model as identified from the
Numerical Renormalization Group. We present analytical formulations for the
subgap bound state energies, the singlet-doublet phase boundary, and the
spectral weights. We show that interactions can reverse the supercurrent across
the trap. We finally work out the spectrum of junction resonators for qubits in
the presence of on-site repulsive electrons and analyze its dependence on
microscopic parameters that may be controlled by fabrication.
