  In this paper, for a family of second-order elliptic equations with rapidly
oscillating periodic coefficients and rapidly oscillating periodic potentials,
we are interested in the $H^1$ convergence rates and the Dirichlet eigenvalues
and bounds of the normal derivatives of Dirichlet eigenfunctions. The $H^1$
convergence rates rely on the Dirichlet correctors and the first-order
corrector for the oscillating potentials. And the bound results rely on an
$O(\varepsilon)$ estimate in $H^1$ for solutions with Dirichlet condition.
