# Self-adjoint extensions for pseudo-differential operators

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?

## 2 Answers

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).

• Thanks, I modified the question with a real principal type symbol $a$. – Bazin Feb 8 at 15:40
• What do you mean by principal symbol? Shubin or classical sense? – mcd Feb 8 at 15:44
• 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 $\Sigma_1$. – Bazin Feb 8 at 15:50
• 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. – mcd Feb 8 at 16:36
• OK, I want ellipticity and the characteristic set to be a manifold, which is not the case of your example $\xi=\pm x$. – Bazin Feb 8 at 19:57

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 $$L^2$$.