Adjoint of differential operator in two variables

Question

I would like to find the adjoint of the operator $$L = x \frac{\partial^{2}}{\partial y^{2}} \frac{\partial }{\partial x}.$$ I know the adjoint is the operator $L^{*}$ such that $$(Lu,v) = (u, L^{*}v)$$ for all $u,v$. But here we are dealing with a differential operator in two variables ($x$ and $y$). How can I find $L^{*}$?

Answer 1

In your formulation the problem is not correctly posed. The operator $L$ is not defined for all $u\in \mathcal{H}=L^{2}(\mathbb{R}^{2},dxdy)$. But it is well-defined on a suitable dense set. Let $p_{x}$ and $p_{y}$ the self-adjoint momentum operators derived from $-i\partial _{x}$ and $% -i\partial _{y}$. Then on the domain of $xp_{x}p_{y}^{2}$ \begin{equation*} L=x\partial _{y}^{2}\partial _{x}=x\partial _{x}\partial _{y}^{2}=-ixp_{x}p_{y}^{2}, \end{equation*} and \begin{equation*} L^{\ast }=ip_{x}xp_{y}^{2}. \end{equation*} For a suitable dense set of $v$'s, such as the compactly supported infinitely differentiable $v$'s,% \begin{equation*} L^{\ast }v=-\partial _{x}x\partial _{y}^{2}v, \end{equation*} which can also be obtained by direct partial integration.