Let $E \to X$ be a smooth vector bundle over a compact Riemannian manifold $X$ and assume that $P:\Gamma(E) \to \Gamma(E)$ is a self-adjoint partial differential operator of order $1$. We think of this operator as an unbounded operator $P:L^2(E) \to L^2(E)$ densely defined on the first order Sobolev space $H^1(E)$. The spectrum $spec(P)$ is always a discrete subset of the real line.
Is $spec(P) \subset \mathbf{R}$ always unbounded from both sides?
In case the order of $P$ is $2$, the claim is wrong for sure (just consider the Laplacian). For example, if $P$ is a Dirac Operator, the claim is always true.
