Consider the differential operator eigenvalue problem $-D^{2}y(x)=E_{n}y(x)$, with boundary periodic conditions $y(x)=y(x+1)$. My question is, if there is a similar problem for the dilation operator $T:= xD$ where $D$ denotes the derivative respect to $x$ which appears in $T^{2}y(x)=E_{n}y(x)$ with the dilation periodic condition, then how can I impose periodic conditions not for the translation group $y(x+1)$ but for the dilation group?
I believe that some periodic conditions involving dilations would be $ y(x)=y(2x)$, but I am not sure.
How can I impose periodic conditions if the lie group is the dilation in 1D?
thanks.