  The orthogonality catastrophe in a Tomonaga-Luttinger liquid with an impurity
is reexamined for the case when the interaction parameter or the dimensionless
conductance is g=1/2. By transforming bosons back to fermions, the Hamiltonian
is reduced to a quadratic form, which allows for explicit calculation of the
overlap integral and the local density of states at the defect site. The
exponent of the orthogonality catastrophe due to a backward scattering center
is found to be 1/8, in agreement with previous studies using different
approaches. The time-dependence of the core-hole Green's function is computed
numerically, which shows a clear crossover from a non-universal short-time
behavior to a universal long-time behavior. The local density of states
vanishes linearly in the low-energy limit at g=1/2.
