Quasinormal modes in the field of a dyon-like dilatonic black hole

Quasinormal modes of massless test scalar field in the background of gravitational field for a non-extremal dilatonic dyonic black hole are explored. The dyon-like black hole solution is considered in the gravitational $4d$ model involving two scalar fields and two 2-forms. It is governed by two 2-dimensional dilatonic coupling vectors $\vec{\lambda}_i$ obeying $\vec{\lambda}_i (\vec{\lambda}_1 + \vec{\lambda}_2) > 0$, $i =1,2$. The first law of black hole thermodynamics is given and the Smarr relation is verified. Quasinormal modes for a massless scalar (test) field in the eikonal approximation are obtained and analysed. These modes depend upon a dimensionless parameter $a$ ($0 < a \leq 2$) which is a function of $\vec{\lambda}_i$. For limiting strong ($a = +0$) and weak ($a = 2$) coupling cases, they coincide with the well-known results for the Schwarzschild and Reissner-Nordstr\"om solutions. It is shown that the Hod conjecture, connecting the damping rate and the Hawking temperature, is satisfied for $0 < a \leq 1$ and all allowed values of parameters.Comment: 13 pages, 5 (double) figures, LaTex. Subsection 3.3 is extended; two typos in formulas (22) and (30) are eliminated; 2 figures and 3 references are added; a paragraph in Sec. 1 and Remark on page 10 are added; several phrases are modified. To be published in EPJ