fa.functional analysis - How hard is it to make a differential operator Hermitian?

Let $M$ be a closed finite-dimensional smooth manifold (over $mathbb R$). Let $C^infty(M) = C^infty(M,mathbb C)$ be the algebra of smooth complex-valued functions on $M$, with the natural complex conjugation $fmapsto bar f$.

Definition: A (real) density $mu$ on $M$ is a section of the trivial real line bundle given in local coordinates by the transition maps $tildemu = bigl| det frac{partial x^i}{partial tilde x^j} bigr| mu$, where $x^i,tilde x^j$ are the different systems of coordinates: if $M$ is oriented, then densities are the same as (real) top-forms. A volume form on $M$ is an everywhere-positive density — this condition is invariant under changes of coordinates.

We remark that if $mu$ is a density, then $int_M mu$ is well-defined: cut $M$ into coordinate patches, integrate each patch, and check that the answer doesn't depend on the choice of coordinates.

A choice of volume form $mu$ determines a Hermitian structure on $C^infty(M)$, via $langle f_1,f_2rangle_mu = int_M {bar f}_1 , f_2 , mu$. A (complex) differential operator $mathcal D: C^infty(M) to C^infty(M)$ is $mu$-Hermitian if $bigllangle f_1, mathcal D[f_2] bigrrangle_mu = bigllangle mathcal D[f_1], f_2 bigrrangle_mu$.

Example: Multiplication by a real-valued function $c$ is Hermitian for any measure. Let $a$ be a Riemannian metric on $M$. Then $sqrt{left|det aright|}$ makes sense as a volume-form on $M$. The Laplace-Beltrami operator on $M$ is given in local coordinates by $Delta_a = -left|det aright|^{-1/2} partial_i a^{ij} left|det aright|^{1/2} partial_j$. It is $sqrt{left|det aright|}$-Hermitian. More generally, let $b$ be any real one-form on $M$. Then I believe that the operator given in local coordinates by:
$$ frac1{sqrt{|det a|}} bigl(sqrt{-1}partial_i + b_ibigr) a^{ij} sqrt{|det a|} bigl( sqrt{-1} partial_j + b_jbigr) $$
is $sqrt{left|det aright|}$-Hermitian.

I believe that the following is true (I've checked it by hand for $M=$ the circle):

Propoposition: Let $a$ be Riemannian metric, $Delta_a$ its Laplace-Beltrami operator, and $mu$ a volume form. Then $Delta_a$ is $mu$-Hermitian if and only if $mu/sqrt{left|det aright|}$ is locally constant.

The if part I said above. Thus, by completing the square, it's straightforward to check whether a second-order differential operator can be made Hermitian with the correct choice of volume form (basically, the factors of $sqrt{-1}$ must be as above). My question is whether you can do this for higher-order differential operators.

Question: Suppose I have a linear differential operator $mathcal D$, with, say, third- or higher derivatives. How do I determine whether there exists a volume-form $mu$ so that $mathcal D$ is $mu$-Hermitian? If such a $mu$ exists, how do I find one? How many are there?

For example, on the circle with the usual volume form, the fourth derivative $partial^4$ is Hermitian, so some of them are. But maybe most of them are not....

Please re-tag as you see fit.