  The behavior of a single impurity in a one-dimensional Luttinger liquid is
numerically investigated by means of the density matrix renormalization group.
By analyzing the finite size scaling behavior of the low energy spectrum, we
confirm the theoretical prediction of Kane and Fisher [Phys. Rev. Lett. {\bf
68}, 1220 (1992)] both for attractive and repulsive interactions. Moreover, we
calculate the exponent of the orthogonality catastrophe, which gives a further
support to the above theoretical prediction.
