A simple derivation of level spacing of quasinormal frequencies for a black hole with a deficit solid angle and quintessence-like matter

In this paper, we investigate analytically the level space of the imaginary part of quasinormal frequencies for a black hole with a deficit solid angle and quintessence-like matter by the Padmanabhan's method \cite{Padmanabhan}. Padmanabhan presented a method to study analytically the imaginary part of quasinormal frequencies for a class of spherically symmetric spacetimes including Schwarzschild-de Sitter black holes which has an evenly spaced structure. The results show that the level space of scalar and gravitational quasinormal frequencies for this kind of black holes only depend on the surface gravity of black-hole horizon in the range of -1 < w < -1/3, respectively . We also extend the range of $w$ to $w \leq -1$, the results of which are similar to that in -1 < w < -1/3 case. Particularly, a black hole with a deficit solid angle in accelerating universe will be a Schwarzschild-de Sitter black hole, fixing $w = -1$ and $\epsilon^2 = 0$. And a black hole with a deficit solid angle in the accelerating universe will be a Schwarzschild black hole,when $\rho_0 = 0$ and $\epsilon^2 = 0$. In this paper, $w$ is the parameter of state equation, $\epsilon^2$ is a parameter relating to a deficit solid angle and $\rho_0$ is the density of static spherically symmetrical quintessence-like matter at $r = 1$.