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 $n\geq 2$ has $λ_1(-αΔ+\operatorname{Ric})\geq 0$ for some $α<\frac{4}{n-1}$ and mean-convex boundary, then it is either isometric to $Σ\times \mathbb{R}_{\geq 0}$ for a closed manifold $Σ$ with nonnegative Ricci curvature or it has no interior ends.