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The behavior of a single impurity in a one-dimensional Luttinger liquid isnumerically investigated by means of the density matrix renormalization group.By analyzing the finite size scaling behavior of the low energy spectrum, weconfirm the theoretical prediction of Kane and Fisher [Phys. Rev. Lett. {\bf68}, 1220 (1992)] both for attractive and repulsive interactions. Moreover, wecalculate the exponent of the orthogonality catastrophe, which gives a furthersupport to the above theoretical prediction.Comment: Latex, 14 Pages (4 PostScript figures), to be appeared in PRB Rapid  Comm., two sentences modified in conclusion and acknowledgemen

Impurity in a Luttinger liquid: A numerical study of the finite-size energy spectrum and of the orthogonality catastrophe exponent