# Closable operator on noncompact manifold

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