same as those of the original process, while the eigenvalues are shifted down by $\Gamma$: $$\lambda^\Gamma_i = \lambda_i-\Gamma,\qquad i\neq 1.$$ This applies, as noted, to all modes except the stationary state, discussed below, which is still such that $\lambda^\Gamma_1=0$.

To determine the left eigenvectors, we act from the left with the original eigenmodes $$\begin{aligned}
\bra{l_i}{\mathcal{L}}^{\Gamma}&=\bra{l_i}\big({\mathcal{L}}+\Gamma\ket{C_0}\bra{-}-\Gamma I\big)\nonumber\\
&=(\lambda_i-\Gamma)\bra{l_i}+\Gamma\left\langle l_i|C_0\right\rangle\bra{-}\nonumber\\
&=(\lambda_i-\Gamma)\left(\bra{l_i}+\frac{\Gamma\left\langle l_i|C_0\right\rangle}{\lambda_i-\Gamma}\bra{-}\right).\nonumber\end{aligned}$$ But by conservation of probability $\bra{-}\mathcal{L}^\Gamma=0$, we also have $$\bra{l_i}{\mathcal{L}}^{\Gamma}=\left(\bra{l_i}+\frac{\Gamma\left\langle l_i|C_0\right\rangle}{\lambda_i-\Gamma}\bra{-}\right){\mathcal{L}}^{\Gamma}$$ for all $i\neq 1$. Thus we see that the new left eigenvectors are given by $$\begin{aligned}
\label{NewLeft}
\bra{l^\Gamma_i}=\bra{l_i}+\frac{\Gamma\left\langle l_i|C_0\right\rangle}{\lambda_i-\Gamma}\bra{-},\qquad i\neq1.\end{aligned}$$ For $i=1$, we have as before $\bra{l_1^\Gamma}=\bra{-}$.

Stationary state {#StatStateResets}
----------------

The stationary state $\ket{P^\Gamma_\text{ss}}$ of the reset process, corresponding to $\ket{r_1^\Gamma}$, is obtained from the results above by noting that the new left and right eigenvectors are orthonormal to each other, so that $\left\langle{l^\Gamma_i}|P^\Gamma_\text{ss}\right\rangle=\delta_{i1}$. Substituting this condition in Eq. , we find $$\left\langle{l_i}|P^\Gamma_\text{ss}\right\rangle=-\frac{\Gamma\left\langle l_i|C_0\right\rangle}{\lambda_i-\Gamma},$$ for $i\neq1$, and thus $$\label{NewSteadyState}
\ket{P^\Gamma_\text{ss}}=\ket{P_\text{ss}}-\sum_{i=2}^{D}\frac{\Gamma\left\langle l_i|C_0\right\rangle}{\lambda_i-\Gamma}\ket{r_i}.$$

We see that in addition to the stationary state of the process without resets, the resetting stationary state contains a contribution of the dynamical modes, weighted according to how significant they are in the evolution of the reset state in the original dynamics (the overlaps $\left\langle l_i|C_0\right\rangle$), 