the particle to hop only to the right with rate $\gamma$, so that the master operator is $${\mathcal{L}}=\gamma\sum_{x=1}^{L}\ket{x+1}\bra{x}-\gamma I.
\label{HoppingMastOp}$$ This operator is translation invariant and can be diagonalized by discrete Fourier transform, with left and right eigenvectors given by $$\ket{r_n}=\frac{1}{L}\sum_{x=1}^{L}e^{i\frac{2\pi n}{L}(x-1)}\ket{x}$$ and $$\bra{l_n}=\sum_{x=1}^{L}e^{-i\frac{2\pi n}{L}(x-1)}\bra{x},$$ respectively, where $n\in\{0,...,L-1\}$. As before, these eigenvectors are normalized such that $\left\langle l_n | r_m \right\rangle = \delta_{nm}$. Moreover, the eigenvalues are given by $$\lambda_n = \gamma(e^{-i2\pi n/L}-1).$$

We now add resets at rate $\Gamma$ onto the reset site $\ket{1}$. The new eigenvalues and eigenvectors can then be calculated exactly using the results of the previous section, with Eq.  leading to $$\left\langle x|P_{\text{ss}}^\Gamma\right\rangle = \frac{\Gamma}{\gamma+\Gamma}S_L(x),$$ where $$S_L(x)=\frac{1}{L}\sum_{n=0}^{L-1}\frac{e^{i\frac{2\pi n}{L}x}}{e^{i\frac{2\pi n}{L}}-\frac{\gamma}{\gamma+\Gamma}}.$$ Considering the sum $S_L(x)$ on different sites, we find $$S_L(x+1)-\frac{\gamma}{\gamma+\Gamma}S_L(x)=\delta_{xL},$$ and so $S_L(x+1)=\gamma/(\gamma+\Gamma)S_L(x)$ when $x\neq L$. Consequently, $$\left\langle x|P_{\text{ss}}^\Gamma\right\rangle = \frac{\Gamma}{\gamma}{\left(\frac{\gamma}{\gamma+\Gamma}\right)}^{x}S_L(1).$$ At this point, rather than explicitly calculate the sum $S_L(1)$, we can just normalise the stationary state to find $$S_L(1)=\frac{1}{1-{\left(\frac{\gamma}{\gamma+\Gamma}\right)}^{L}},$$ thus giving $$\label{NewHoppingStationary}
P_\text{ss}^\Gamma(x)=\left\langle x|P_{\text{ss}}^\Gamma\right\rangle = \frac{\Gamma}{\gamma+\Gamma}\frac{1}{1-{\left(\frac{\gamma}{\gamma+\Gamma}\right)}^{L}}{\left(\frac{\gamma}{\gamma+\Gamma}\right)}^{x-1}.$$

This result is interesting because, while the infinite-size limit of the initial model neither has a stationary state or the ability to reach one (the spectral gap tends to zero), the infinite-size limit of the model with resets 