LOCAL PERTURBATION IN A TOMONAGA-LUTTINGER LIQUID AT G=1 2 - ORTHOGONALITY CATASTROPHE, FERMI-EDGE SINGULARITY, AND LOCAL-DENSITY OF STATES/

The orthogonality catastrophe in a Tomonaga-Luttinger liquid with an i mpurity is reexamined for the case when the interaction parameter or t he dimensionless conductance is g = 1/2. By transforming bosons back t o fermions, the Hamiltonian is reduced to a quadratic form, which allo ws for explicit calculation of the overlap integral and the local dens ity of states at the defect site. The exponent of the orthogonality ca tastrophe due to a backward scattering center is found to be 1/8, in a greement with previous studies using different approaches. The time de pendence of the core-hole Green's function is computed numerically, wh ich shows a clear crossover from a nonuniversal 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.