this section our formula for the stationary state of the reset process for two exactly-solvable models. The applicability of this formula is obviously limited by the fact that it requires the full spectrum of the reset-free process. For this reason, we expect it to be more useful for approximating the stationary state than for calculating that state exactly, either by truncating the sum involved to a limited number of modes or by expanding the sum perturbatively in $\Gamma$. Moreover, while exact results can be hard to find, the formula can be useful numerically when applied to processes in which the resets break symmetries of the original, reset-free process. Such symmetries can indeed be used to diagonalise the original process for system sizes much larger than would otherwise be possible, with the resulting spectrum then being used in to derive the stationary state with resets. This is demonstrated later in the context of adding resets to a closed quantum system breaking time-reversal symmetry.

Totally asymmetric random walk {#AsymHop}
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The first model that we consider is a particle hopping on a one-dimensional lattice of length $L$ with periodic boundary conditions, so the states are $\ket{x}$ with $x\in\{1,2,\ldots,L\}$ and $\ket{L+1}=\ket{1}$. We take 