Spectral rigidity of manifolds with Ricci bounded below and maximal bottom spectrum

We investigate the spectrum of the Laplacian on complete, non-compact
manifolds $M^n$ whose Ricci curvature satisfies $\mathrm{Ric} \geq
-(n-1)\mathrm{H}(r)$, for some continuous, non-increasing $\mathrm{H}$ with
$\mathrm{H}-1 \in L^1(\infty)$. We prove that if the bottom spectrum attains
the maximal value $\frac{(n-1)^2}{4}$ compatible with the curvature bound, then
the spectrum of $M$ coincides with that of hyperbolic space $\mathbb{H}^n$,
namely, $\sigma(M) = \left[ \frac{(n-1)^2}{4}, \infty \right)$. The result can
be localized to an end $E$ with infinite volume.