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



first-order linear differential equation with boundary conditions
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 $cx^{iE\_n-1/2}$ and your condition gives $p^{iE\_n-1/2}=1$. This is satisfied if and only if it exists some integer $n\in\mathbb{Z}$ such that $(iE\_n-1/2)\ln p=2in\pi$. 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
$$E=-i/2\qquad f(x)=c$$


