I use the following def. of the Fourier transform $$ f(\vec{x}) = \frac{1}{(2\pi)^2}\int_{\mathbb{R}_2} F(\vec{k}) e^{i\vec{k}\cdot\vec{x}}dk^2 $$ and also $$ \Delta_{\vec{x}}\int_{\mathbb{R}_2} F(\vec{k}) e^{i\vec{k}\cdot\vec{x}}dk^2 = - \int_{\mathbb{R}_2}||\vec{k}||^2 F(\vec{k}) e^{i\vec{k}\cdot\vec{x}}dk^2 $$
Now I want to proof: $$ ||(I-a\Delta_{\vec{x}})^{l/2}f(\vec{x})||^2 = \big|\big|(I+a||\vec{k}||^2)^{l/2} F(\vec{k})\big|\big|^2 $$ However, I am not sure if there is some factor $2\pi$ missing...
For the proof I have to use the Fourier representation of the dirac-distribution $$ \delta(\vec{k}) = \frac{1}{(2\pi)^\text{?}}\int_{\mathbb{R}_2} e^{i\vec{k}\cdot\vec{x}} dx^2 $$ But I am not sure about the power of $1/2\pi$. If you could tell me what power is correct, I could determine the possibly missing factor in the proof...
Can you help? (note that I have to use the non-unitary definition of the FT... this is what causes all the trouble)