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 $\mathrm{R}\mathrm{i}\mathrm{c}\ge -(n-1)\mathrm{H}(r)$, for some continuous, non-increasing $\mathrm{H}$ with $\mathrm{H}-1\in L^1(\mathrm{\infty })$. 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, $\sigma (M)=[\frac{(n-1{)}^2}{4},\mathrm{\infty })$. The result can be localized to an end $E$ with infinite volume.