Write $x=g^\eta$. Applying $D_x$ and then changing the summation variable from $k$ to $\ell=k-\alpha\eta$ gives
$$
\begin{aligned}
D_x|\alpha\rangle|\chi_\beta\rangle
&=\frac1{\sqrt N}\sum_k
\omega^{\beta k}|\alpha\rangle|g^{k-\alpha\eta}\rangle\\
&=\omega^{\alpha\beta\eta}
|\alpha\rangle|\chi_\beta\rangle.
\end{aligned}
$$
Hence this state is an <eigenvector> of $D_x$ with <eigenvalue> $\boxed{\omega^{\alpha\beta\eta}}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 324 / 1 / c / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-324/1/c/ii/solution