Let us define $$F_n=\int f(z) |He_n(z)|^2 \, dz \, dz^*$$ is there any type of function $f$ could make that $F_n=0$ for $n\geq 2$ and $F_n>0$ for $n<2?$
$He_n(x)=2^{-\frac{n}{2}}H_n\left(\frac{x}{\sqrt{2}}\right)$ and $H_n$ is the usual physicist's Hermite polynomial.