  A theorem of Llarull says that if a smooth metric $g$ on the $n$-sphere
$\mathbb{S}^n$ is bounded below by the standard round metric and the scalar
curvature $R_g$ of $g$ is bounded below by $n (n - 1)$, then the metric $g$
must be the standard round metric. We prove a spectral Llarull theorem by
replacing the bound $R_g \geq n (n - 1)$ by a lower bound on the first
eigenvalue of an elliptic operator involving the Laplacian and the scalar
curvature $R_g$. We utilize two methods: spinor and spacetime harmonic
function.
