Let $M$ be a smooth manifold, $E$ be a Clifford bundle over $M$ and $D$ be the associated Dirac operator on $E$.
If $M$ is compact then $D$ is a Fredholm operator, since it is elliptic.
I'm wondering whether there are any easy examples of non-compact manifolds $M$ and bundles $E$ where one can show that $D$ is not Fredholm? I would assume $M=\mathbb{R}$ might be one such manifold, but I wasn't sure how to go about proving that the Dirac operator in this case is not Fredholm.
Thanks.