It is extremely difficult to obtain multiplicities of tightly clustered eigenvalues over domains perturbated from a domain with multiple eigenvalues, since it requires infinitely tight bounds of the eigenvalues in the neighborhood of the domain with multiple eigenvalues. In this study, we propose a difference quotient formula that describes the relationship between eigenvalues and eigenfunctions over the original and perturbated domains. By tracking the perturbation behavior of the eigenspace using the error estimation of eigenspaces recently developed by the second author, we established a guaranteed computation method for the difference quotient.