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The class $\Sigma^1$ of symbols on $\mathbb R^{2n}$ is made with $C^\infty$ functions $a$ of $X=(x,\xi)\in \mathbb R^n\times\mathbb R^n$ such that $$ \vert\partial_X^\alpha a\vert\le C_\alpha(1+\vert X\vert)^{2-\vert \alpha\vert}. $$ Assuming that $a\in\Sigma^1$ is real-valued principal type and denoting by $A$ its Weyl quantization, using the fact that $A$ is continuous on the Schwartz space $\mathscr S(\mathbb R^n)$ and on its dual (tempered distributions) $\mathscr S'(\mathbb R^n)$, we may define the maximal extension $H$ of $A$ with the domain $$ D(H)=\{u\in L^2(\mathbb R^n), Au \in L^2(\mathbb R^n) \}. $$
Then I claim that $H$ is self-adjoint. I believe that it is well-known and I am looking for a reference in the literature.

A related question is the same problem for first-order pseudo-differential operators on a compact manifold without boundary $\mathcal M$ (equipped with a smooth density): let $A$ be a first-order pseudo-differential operator on $\mathcal M$ (I do not want to assume ellipticity, but I know that $A$ is continuous on $C^\infty(\mathcal M)$ and on the distributions on $\mathcal M$) and assume that $A$ is symmetric, that is such that for $\phi, \psi\in C^\infty(\mathcal M)$ $\langle A\phi,\psi \rangle=\langle \phi,A\psi \rangle_{L^2(\mathcal M)}$. Then consider the maximal extension $H$ of $A$ with $$ D(H)=\{u\in L^2(\mathcal M), Au \in L^2(\mathcal M) \}. $$ Then $H$ is self-adjoint. Is it true and well-known?

Last but not least, dropping the compactness assumption on $\mathcal M$ in the second question above, assuming that $A$ is properly supported, can I get the same result?

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The harmonic oscillator with non-positive coupling constant, i.e., $H_0 f = -f''(x) + cx^2f(x)$, with $c \in \mathbb{C} \setminus \mathbb{R}$, is non-self-adjoint (see the book in preparation by Sjöstrand (http://sjostrand.perso.math.cnrs.fr/161014bok.pdf)). This does not answer the question since $c = -1$ is not allowed (and gives a symmetric operator).

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  • $\begingroup$ Thanks, I modified the question with a real principal type symbol $a$. $\endgroup$ – Bazin Feb 8 at 15:40
  • $\begingroup$ What do you mean by principal symbol? Shubin or classical sense? $\endgroup$ – mcd Feb 8 at 15:44
  • $\begingroup$ I mean that $a=a_1+a_0$ with $a_j\in \Sigma^j$ is such that $\vert \nabla a_1\vert^2$ is elliptic in $\Sigma_1$. $\endgroup$ – Bazin Feb 8 at 15:50
  • $\begingroup$ In my example with $a_1(x,\xi) = \xi^2 - x^2$ and $a_0 = 0$, we have that $|\nabla p_2(x,\xi)|^2 = 4(\xi^2 + x^2)$, which is elliptic. $\endgroup$ – mcd Feb 8 at 16:36
  • $\begingroup$ OK, I want ellipticity and the characteristic set to be a manifold, which is not the case of your example $\xi=\pm x$. $\endgroup$ – Bazin Feb 8 at 19:57
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The affirmative answer to your second question is indeed well-known. For example, in

Commun. Math. Phys. 308, 325–364 (2011), Lemma A.1 on page 358,

Frédéric Faure and Johannes Sjöstrand prove that every PDO of order 1 on a closed manifold has a unique closed extension in $L^2$.

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