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[en] The interplay of an Anderson quantum impurity with topological surface modes is studied. We find that the quantum impurity scattering can locally destroy the Dirac electron spectra by creating a significant resonance exactly at the Dirac point, in stark contrast to the case of classic impurities. When an external magnetic field is applied to the topological insulator (TI) surfaces, a bound state is found either at the gap edges or within the gap. We discuss the coexistence of the Kondo resonance and the bound state and their effect on TI local density of states. - Highlights: • A resonance at the Dirac point is found, differing from classic impurity theory. • A magnetic field-induced bound state is found within the energy gap. • Impurity Kondo resonance can cause corresponding signatures in the LDOS of TIs. • The results can be tuned by a gate voltage or a chemical potential.