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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 scalarcurvature $R_g$ 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 byreplacing the bound $R_g \geq n (n - 1)$ by a lower bound on the firsteigenvalue of an elliptic operator involving the Laplacian and the scalarcurvature $R_g$. We utilize two methods: spinor and spacetime harmonicfunction.

Spectral constant rigidity of warped product metrics