  In this paper, we derive an asymptotic error expansion for the eigenvalue
approximations by the lowest order Raviart-Thomas mixed finite element method
for the general second order elliptic eigenvalue problems. Extrapolation based
on such an expansion is applied to improve the accuracy of the eigenvalue
approximations. Furthermore, we also prove the superclose property between the
finite element projection with the finite element approximation of the
eigenvalue problems by mixed finite element methods. In order to prove the full
order of the eigenvalue extrapolation, we first propose "the auxiliary equation
method". The result of this paper provides a general procedure to produce an
asymptotic expansions for eigenvalue approximations by mixed finite elements.
