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