# Spectrum of the Laplace-Beltrami operator on $L^p$: where is it?

On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ depending on sign convention. I know that under various curvature assumptions one can show that the $L^p$-spectrum is equal to the $L^2$-spectrum for $p \neq 2$.

Without making any geometric assumptions on $M$, is it possible to conclude that the $L^p$-spectrum of $\Delta$ is a subset of $\mathbb{R}$? If not, is there anything we can say about the $L^p$-spectrum without having to make any additional assumptions?

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