Let us take a compact Riemannian manifold $M$. Let us define $-\Delta:H^1(M) \to H^{-1}(M)$ by $$\langle -\Delta u, v \rangle = \int_M \nabla u \nabla v$$ and $-\tilde \Delta:L^2(0,T;H^1(M)) \to L^2(0,T;H^{-1}(M))$ by $$\langle -\tilde \Delta f, g \rangle = \int_0^T\int_M \nabla f(t) \nabla g(t).$$
Is there some relationship between the eigenfunctions and eigenvalues of these operators? I think the eigenvalues of $-\tilde \Delta$ evaluated at a time $t$ are the eigenvalues of $-\Delta$. Right??