$$0=\Gamma\ket{C_0}\bra{C_0}-\Gamma\ket{C_0}\bra{C_0}$$ to ${\mathcal{L}}^\Gamma$ to obtain the simpler form $$\begin{aligned}
{\mathcal{L}}^\Gamma&={\mathcal{L}}+\Gamma\sum_{C} \ket{C_0}\bra{C}-\Gamma\sum_{C}\ket{C}\bra{C}\nonumber\\
&={\mathcal{L}}+\Gamma\ket{C_0}\bra{-}-\Gamma I,
\label{LR}\end{aligned}$$ where $I$ is the identity operator. In this form, it is clear that the reset adds transitions from all states to $C_0$, contributing to an extra escape rate $\Gamma$ in the diagonal, which keeps the conservation condition $\bra{-}{\mathcal{L}}^\Gamma=0$.

The dynamics generated by is arguably the simplest form of reset process. As defined, reset events are Poissonian with rate $\Gamma$, which is independent of the state of the system at the time of the reset.

Results {#SecRes}
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We study in this section the spectral properties of the reset process, deriving the new eigenvalues, left eigenvectors and right eigenvectors of ${\mathcal{L}}^\Gamma$ in terms of those of ${\mathcal{L}}$. The results are then used to obtain spectral representations of the stationary state and current of the reset process, and to discuss the effect of resets on metastable states.

Eigenvalues and eigenvectors {#ResetSpec}
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We begin our analysis with the right eigenvectors by noting that, due to conservation of probability and the fact that $\left\langle -|r_i \right\rangle=0$ for all $i\neq1$, we have $$\begin{aligned}
{\mathcal{L}}^\Gamma\ket{r_i} &=\big({\mathcal{L}}+\Gamma\ket{C_0}\bra{-}-\Gamma I\big)\ket{r_i}\nonumber\\
&=(\lambda_i-\Gamma)\ket{r_i}\end{aligned}$$ for $i\neq 1$. Consequently, the right eigenvectors of the reset process are the 