  Double-exchange system is investigated by the dynamical mean-field theory. We
show that the chemical potential shifts as a function of temperature and
magnetization, which is anomalously large. We also discuss the influences of
dynamic Jahn-Teller effect to the shift of the chemical potential. Measurement
of the shift of the chemical potential casts a constraint to theoretical
approaches for the magnetoresistance phenomena in ($R$,$A$)MnO$_3$ such as
double-exchange effects and dynamic Jahn-Teller effects. We also propose a
method to measure the shift of $\mu$.
