Discreteness of Dyonic Dilaton Black Holes

We show that there are two classes of solutions describing static spherically symmetric dyonic dilaton black holes with two nonsingular horizons. The first class includes only the already known solutions that exist for a few special values of the dilaton coupling constant. Solutions in the second class have essentially different properties. They exist for continuously varying values of the dilaton coupling constant but arise only for discrete values of the dilaton field at the horizon. For each given value of the dilaton coupling constant, there can exist several such solutions differing by the number of zeros of the shifted dilaton function in the subhorizon region and separating the domains of singular solutions.