Classical completeness and quantum completeness for formally self-adjoint differential operators

Let $X$ be a closed smooth manifold equipped with a smooth density $|dx|$, and let $P$ be a formally self-adjoint differential operator on $C^fty(X)$. Write $p$ for the symbol of $P$, and call $P$ essentially self-adjoint (ESA), or quantum complete, when it has a unique self-adjoint extension.

Classical–quantum completeness conjecture. The Hamiltonian flow of $p$ is complete if and only if $P$ is essentially self-adjoint.

This proposes an equivalence between completeness of the classical Hamiltonian dynamics and quantum completeness of the associated differential operator on a closed manifold.