A splitting theorem for manifolds with spectral nonnegative Ricci curvature and mean-convex boundary

We prove a splitting theorem for a smooth noncompact manifold with (possibly noncompact) boundary. We show that if a noncompact manifold of dimension $n\ge 2$ has ${\lambda }_1(-\alpha \mathrm{\Delta }+\mathrm{Ric})\ge 0$ for some $\alpha <\frac{4}{n-1}$ and mean-convex boundary, then it is either isometric to $\mathrm{\Sigma }\times {\mathbb{R}}_{\ge 0}$ for a closed manifold $\mathrm{\Sigma }$ with nonnegative Ricci curvature or it has no interior ends.