I found this discussion on Wikipedia regarding the self-adjointnes of the momentum operator on a bounded domain. https://en.m.wikipedia.org/wiki/Self-adjoint_operator#Boundary_conditions
The problem discussed there is similar to one I am dealing with, except for the fact that there is an extra dimension in play. Concretely, I have a symmetric operator of the form:
$$T=A(x)\partial_x+B(x)$$
which should act on the set of square integrable functions $f:[0,1]\times [0,1]\rightarrow \mathbb{C}$ with doubly periodic boundary conditions: $f(0,y)=f(1,y)$ and $f(x,0)=f(x,1)$ for all $x,y\in [0,1]$. Is this operator self-adjoint for the right choice of the domain, like the momentum operator, or are there some complications arising in this case?