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Let $L$ be a differential operator in $L^2(M, dvol)$ wrt to a Riemannian volume form (say). Let us call it quantum complete if it is essentially-self-adjoint. Consider $H$ - the symbol of $L$. It is a function on a cotangent bundle of $M$, i.e. a Hamiltonian. This Hamiltonian defines a Hamiltonian flow on $T^\ast M$.

I am looking for a good reference which discusses the relation between the quantum completeness of $L$ and the completeness of the Hamiltonian flow of $H$. Some information is contained in Reed and Simon but they seem to consider only the case of $M=(0, \infty)$.

Particular questions I am interested in:

1) Sometimes the Hamiltonian flow blows up in finite time but the corresponding $L$ is complete. What is the physical meaning of this?

2) Is it possible to use the completeness of the Hamiltonian flow to prove that $L$ is ess. self.adjoint? For example, is it possible to apply this logic to prove that the Laplace-Beltrami operator is ess. self-adjoint on a geodesically complete manifold (which is a Hamiltonian flow of its symbol)?

Thanks,

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I discuss these sort of issues in this blog post of mine (particularly in Section 4).

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