  Recent investigations revealed that the near horizon Hamiltonian of a
massless, chargeless outgoing particle, for its particular motion in static as
well as stationary black holes, is effectively $\sim xp$ kind. This is unstable
by nature and has the potential to explain a few interesting physical
phenomena. From the path integral kernel, we first calculate the density of
states. Also, following the idea of [Phys. Rev. D 85, 025011 (2012)] here, in
the vicinity of the horizon, we calculate the effective path corresponding to
its Schrodinger version of Hamiltonian through the path integral approach. The
latter result appears to be complex in nature and carries the information of
escaping the probability of the particle through the horizon. In both ways, we
identify the correct expression of Hawking temperature. Moreover, here we
successfully extend the complex path approach to a more general black hole like
Kerr spacetime. We feel that such a complex path is an outcome of the nature of
near horizon instability provided by the horizon and, therefore, once again
bolstered the fact that the thermalization mechanism of the horizon may be
explained through the aforesaid local instability.
