Thermal hair of a quantum black hole

We reexamine the possibility which has been argued since just after the discovery of Hawking radiation: the statistical explanation of Bekenstein-Hawking entropy by counting quasibounded modes of thermal fluctuation in two-dimensional black hole spacetime. While string theory has made much progress recently, it may be still interesting to study black hole entropy with field theories. The black hole concerned is quantum in the sense that it is in thermal equilibrium with its Hawking radiation. It is shown that the fluctuation around such a black hole obeys a wave equation with a potential whose peaks are located near the black hole and which is caused by quantum effects. We can construct models in which the potential in the above sense has several positive peaks and there are quasibounded modes confined between these peaks. This suggests that these modes contribute to black hole entropy. However, it is shown that the entropy associated with these modes does not obey the ordinary area law. We can call these modes additional thermal hair of the quantum black hole. Therefore the situation for the possibility is more difficult and we might find other ways to explain the entropy. From recent progress counting the number of states in string theory, it becomes more convincing that we expect Bekenstein-Hawking entropy for the Schwarzschild black hole will be explained exactly from the string theoretical point of view.