let be the differential equation
$ ixDf(x)if(x)/2= E_{n}f(x) $
with the boundary conditions $ f(x)=f(p^{k}x) $ for 'p' prime and $k=...,2,1,0,1,2,...$
is this possible to solve this eigenvalue problem ?? thanks
let be the differential equation $ ixDf(x)if(x)/2= E_{n}f(x) $ with the boundary conditions $ f(x)=f(p^{k}x) $ for 'p' prime and $k=...,2,1,0,1,2,...$ is this possible to solve this eigenvalue problem ?? thanks 


As I understand the question, you first fix $p$, and then you search the solutions of $$−ixDf(x)−if(x)/2=E_nf(x)$$ which satisfy $f(px)=f(x)$ (by reccurence, your $p^k$ conditions is automatically satisfied). So we solve the differential equation and we find $cx^{iE_n1/2}$ and your condition gives $p^{iE_n1/2}=1$. This is satisfied if and only if it exists some integer $n\in\mathbb{Z}$ such that $(iE_n1/2)\ln p=2in\pi$. So the possible values of $E_n$ and then eigenfunctions are $$E_n=\frac{2in\pi}{\ln p}i/2\qquad f_n(x)=ce^{\frac{2in\pi\ln x}{\ln p}}$$ If you intend not to fix first the value of $p$, then we need to consider only the solutions valid for all prime $p$. Then only solution is then for $n=0\;$ which gives $$E=i/2\qquad f(x)=c$$ 

