Influence of the back reaction of the Hawking radiation upon black hole quasinormal modes

We consider the BTZ black hole surrounded by the conformal scalar field. Within general relativity, the resonant quasinormal (QN) modes dominate in the response of a black hole to external perturbations. At the same time, the metric of an evaporating black hole is affected by the Hawking radiation. We estimate the shift in the quasinormal spectrum of the BTZ black hole stipulated by the back reaction of the Hawking radiation. For the case of the $\mathrm{}(2+1)\mathrm{}$-dimensional black hole the corrected (by $\sim ħ)$ metric is an exact solution [C. Martines and J. Zanelli, Phys. Rev. D 55, 3642 (1997)]. In addition, in this case quantum corrections come only from matter fields and not from graviton loops, that is, one can solve the problem of influence of the back reaction upon the QN ringing self-consistently. The dominant contribution to the corrections to the QNMs is simply a shift of ${\omega }^2$ proportional to $-(\Lambda {/M)}^{3/2}{(4L}^2+M)ħ.$ It is negligible for large black holes but essential for small ones, giving rise to considerable increasing of the quality factor. Thus, the small evaporating black hole is expected to be a much better oscillator than a large one.