Looking for a paper on (formally) self-adjoint differential operators

Question

This is a long shot, but I've about lost my mind over this. About a year ago, I came across a paper published in the last 20-30 years (as it was neatly typeset in modern $\rm\LaTeX$ styles) that extended Kodaira's work on self-adjoint differential operators of even order on $L^2(\mathbb{R})$ which itself effectively generalizes Sturm-Liouville theory. In my infinite wisdom, I did not save the paper which is proving to be a very painful learning experience..

This paper shares some commonality with my research, but unfortunately, some of the terminology is not standardized exactly and littered with lots of very tangential work making it difficult to find again. The paper considers differential operators on $L^2(\mathbb{R})$ of the form

$$ D = \sum_{i=0}^n p_j(x) \frac{d^j}{dx^j} $$

and their formal adjoints

$$ D^+ = \sum_{i=0}^n (-1)^n \frac{d^j}{dx^j} {p_j(x)}. $$

The author specifically mentioning the limitation of Kodaira's work to the even order case. The paper demonstrates combinatorial identities that the $p_j$ must solve for these differential operators to be formally self-adjoint. If I recall correctly - and this is where the hook with my research lies - it considers the special case that the $p_j$ are all powers of $x$ (relating to the dilation/scaling operator) to be able to say more definitive statements about the differential operators than they would be able to otherwise due to generality.

Most of the work in this direction occurred in the 1950s and 1960s, so this paper stood out to me quite a bit. Levinson's paper in 1953ish is fairly similar to the paper I have in mind, but it misses the "power of $x$" bit. Zettl's survey is somewhat close, but that isn't it either.

Any direction would be much appreciated!

Answer 1

Let me start by altering a bit your notations: we consider a differential operator $P$ defined by $$ P=\sum_{1\le j\le n}p_j(x) D^j, \quad D=-i\frac{d}{dx}. $$ The formal adjoint is (there is a typo in your statement with a $(-1)^n$ which should be replaced by $(-1)^j$) $$ P^*=\sum_{1\le j\le n} D^j \overline{p_j(x)}. $$ Instead of fighting to get conditions on the $p_j$ to get $P=P^*$, I propose to use the Weyl quantization, introduced by Hermann Weyl in 1926: let us define $$ (Pv)(x)=\iint \sum_{0\le j\le n} q_j\bigl(\frac{x+y}{2}\bigr)\xi^j e^{i(x-y)\xi} v(y) dy d\xi(2π)^{-n}. $$ Then an iff condition to get formal self-adjointness is that $$\forall (x,\xi)\in \mathbb R^2, \quad \sum_{0\le j\le n} q_j(x)\xi^j\in \mathbb R, $$ which in that case means $ \forall x\in \mathbb R, q_j(x)\in \mathbb R. $ You can of course express the Weyl symbol $q(x,\xi)=\sum_{1\le j\le n}q_j(x)\xi^j$ of $P$ in terms of its standard symbol $$ p(x,\xi)=\sum_{1\le j\le n}p_j(x)\xi^j, $$ by the formula $ q=\exp(-\frac{i}{2}D_xD_\xi)p, \quad D_x=-i\frac{d}{dx}, D_\xi=-i\frac{d}{d\xi}. $ As a consequence, an iff condition for formal selfadjointness is $$\forall (x,\xi)\in \mathbb R^2, \quad \bigl(\exp(-\frac{i}{2}D_xD_\xi)p\bigr)(x,\xi)\in \mathbb R. $$