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Let $M$ be a non-compact complete manifold. Suppose that $L$ is an elliptic operator, e.g. Schrodigner operator.

We know that if the domain of the adjoint $L^*$ is dense in $L^2$, we have that $L$ is closable.

Q If we drop the condition that $C^\infty_c$ is dense in $L^2$, can we still have $L$ is closable, e.g. manifold with unbounded geometry. f

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