Quasinormal frequencies and thermodynamic instabilities for the stationary axisymmetric Einstein-Maxwell dilaton-axion black hole

The massless scalar quasinormal frequencies of a stationary axisymmetric Einstein-Maxwell dilaton-axion (EMDA) black hole are investigated by using Leaver's continued fraction method. It is shown that in the complex plane the frequencies move counterclockwise and get a spiral-like shape as the angular momentum per unit mass a increases to its extremal value or the dilaton D decreases to its extremal value for the rotating black hole. However, for the non-rotating Garfinkle-Horowitz-Strominger dilaton (GHSD) black hole, the dilaton parameter D, which is related to the electric charge of this EMDA black hole, cannot make the frequencies spire in the complex w plane, which is qualitatively different from the charge of the Reissner-Nordstrom (RN) black hole. The so-call "Spiral-like Criterion" is obtained and it points out that the frequencies won't spire in the complex w plane if the heat capacity for the considered black hole is always negative and vice versa. The most interesting outcome of our calculation is that the critical point, at which the imaginary part of the wave function related to time-dependent part (e-(iwt)) begins to oscillate obviously for the given quantum number, is just the second order phase transition point of Davies. The fact seems to imply that there is some relation between the dynamical evolution and thermodynamic instabilities for the black hole.