in this paper [on arxiv](http://arxiv.org/pdf/1207.4985v1.pdf) in equation 27, two operators 


$$A\_m^\* = (1-x^2)^{\frac{1}{2}} \frac{d}{dx} + \frac{mx}{\sqrt{1-x^2}}$$


and $$A\_m = - \frac{d}{dx}(1-x^2)^{\frac{1}{2}} + \frac{mx}{\sqrt{1-x^2}}$$


belonging to Legendre's ODE are defined. Unfortunately, a spectral analysis of these two operators is missing. 


Therefore, I was wondering if anybody here knows a domain $D(A\_m) \subset L^2([-1,1])$ such that $A\_m$ is a closed operator with adjoint $A\_m^\*$



Proper domain for operators
Minimal and maximal operators will work fine: Define an operator on $C\_0^{\infty}(-1,1)$, using the differential expression of $A$, and then take the operator closure to obtain $A$. Then $A^\*$ will be the corresponding maximal operator, that is,
$$
D(A^\*) = \{ u\in L^2(-1,1)\cap AC(-1,1): (1-x^2)^{1/2}Du + v(x)u \in L^2(-1,1) \} ;
$$
here I write $v$ for the function from the multiplication operator part.


