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Consider the following operator: $ -i\hbar x\frac{df(x)}{dx}-i\hbar \frac{f(x)}{2}=E_{n}f(x)$

What kind of boundary conditions can I enforce? I have tried to find a function such that for every integer $n$ I get $ f(nx)=f(x) $, but I only get that $ f(x) $ must be a constant.

Of course I know how to solve it to get the solution $ f(x)= \frac{C}{x^{1/2-iE_{n}}} $

or perhaps to set the conditions $ F(nx)=F(x) $ for every integer $n$ with given

$ F(x)= \sum_{p} \sum_{k=-\infty}^{\infty}f(p^{k}x) $ with $ f(x) $ solving the above equation and 'p' runs over the primes $ p=2,3,5,7,11,... $

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No offense, but this sort of sounds like a homework problem. Is it? –  Colin Fredericks Jun 1 '12 at 18:40
    
no, it isn't i have a degree on physics :) (although i am currently unemployed) i was looking for a solution of this strange problem :S –  Jose Javier Garcia Jun 1 '12 at 19:02
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@JoseJavierGarcia: FYI, it doesn't matter if it was assigned as actual homework or not to qualify as homework under the SE rules. –  JamalS Dec 18 at 11:04

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