  We consider a massless, minimally-coupled quantum scalar field on a
Reissner-Nordstrom black hole background, and we study the leading asymptotic
behavior of the expectation value of the stress energy tensor operator
$\langle\hat{T}_{\mu\nu}\rangle_{ren}$ and of
$\langle\hat{\Phi}^{2}\rangle_{ren}$ near the inner horizon, in both the Unruh
and the Hartle-Hawking quantum states. We find that the coefficients of the
expected leading-order divergences of these expectation values vanish,
indicating that the modifications of the classical geometry due to quantum
vacuum effects might be weaker than expected. In addition, we calculate the
leading-order divergences of $\langle\hat{T}_{\mu\nu}\rangle_{ren}$ and of
$\langle\hat{\Phi}^{2}\rangle_{ren}$ in the Boulware state near the outer
(event) horizon, and we obtain analytical expressions that correspond to
previous numerical results.
