If $(M,g)$ is a Riemannian manifold and $\Delta$ is the Laplace-Beltrami (negatively defined) operator, is it possible to describe the class of smooth potentials $V :M \to \Bbb R$ that make the heat operator $\partial_t - \Delta + V$ on $(0,\infty) \times M$ hypoelliptic? If this is too complicated (as I suspect it is), then is it possible to impose reasonable sufficient conditions on $(M,g)$ and $V$ in order to obtain this hypoellipticity?
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1$\begingroup$ Inuitively, how could such an operator not be hypoelliptic, given that the leading term is elliptic? If $V$ is very wild, you might have problems with existence of solutions, but it seems to me that if they exist they should be smooth. $\endgroup$– Nate EldredgeCommented Feb 22, 2017 at 15:41
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$\begingroup$ @NateEldredge: $\Delta$ is elliptic on $M$, but not on $(0,\infty) \times M$. Also, the "true" Schrödinger operator $\Bbb i \partial_t + \Delta$ is known to not be hypoelliptic. Or maybe I'm just not following you. $\endgroup$– Alex M.Commented Feb 22, 2017 at 15:54
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