Given the operator $A = (X\frac{d}{dx}+2)$, where $X$ is a linear operator, how can I find the eigenfunction of $A$ corresponding to a zero eigenvalue?
In general, this is just a matter of solving the differential equation for $AF(x) = 0$, however, in this case, that leaves me with the differential equation $X \frac{d}{dx} F(x) + 2 F(x) = 0$, and I'm just not really sure what to do with that $X$ operator.
Anyone out there that can get me past this step?
Your operator X may be thought of as a constant, since it does nothing on "numbers", which include the parameter x, so it commutes with everything just like a constant.
It is then evident that $$ X\frac{d}{dx} F(x,X)= -2 F(x,X) $$ is solved by $$ F(x,X)=c e^{{-2x}X^{-1}} , $$ where $X^{-1}$ is the inverse operator to X.