From Quasinormal Modes to Logarithmic Corrections: Role of Geometric Deformation in Dilaton Black Hole Thermodynamics

We investigate the quasinormal mode (QNM) spectrum of a four–dimensional dilaton black hole with a deformed angular geometry [Formula: see text], where [Formula: see text] and [Formula: see text], and its connection to logarithmic corrections in black hole entropy. By analyzing scalar perturbations in the background spacetime, we derive the effective potential for the Schrödinger–like wave equation and compute the complex QNM frequencies using the third–order WKB approximation. The parameter [Formula: see text] significantly influences the shape of the potential and the damping rate of perturbations, leading to distinct dynamical behaviors for generic [Formula: see text], [Formula: see text], and [Formula: see text]. We propose a conjectural relation between the asymptotic QNM frequencies and the logarithmic correction coefficient [Formula: see text] in [Formula: see text], given by [Formula: see text], which is supported by WKB analysis and numerical fits, and reduces to the known value [Formula: see text] when [Formula: see text]. For [Formula: see text] and [Formula: see text] (which lie in the physical range [Formula: see text] but outside the domain of validity of the asymptotic formula), the system exhibits critical or unstable behavior. These results demonstrate that the geometric deformation parameter [Formula: see text] acts as a fundamental control parameter in the phase structure and quantum thermodynamics of dilaton black holes. We further verify that in the limit [Formula: see text], [Formula: see text], the solution reduces to the Reissner–Nordström–AdS black hole, confirming the model as a consistent deformation of classical black holes.