how resets can accelerate or even induce relaxation to a stationary state by opening a spectral gap, and how non-zero stationary currents can be created without having complex eigenvalues in the spectrum of the generator. The eigenvalue result can also be used within the spectral theory of metastability [@GaveauSchulman1987Meta; @Gaveau1998; @Gaveau2006] to demonstrate that weak resetting can modify the weighting of metastable states without modifying those states as such.

While resets have been extensively considered in classical nonequilibrium physics, a relatively unexplored area is the addition of resets to quantum systems, either closed or interacting with an environment. This case has been considered recently for a quantum walker subjected to continuous measurements on a particular site, resulting in a random collapse of the wavefunction followed by an evolution starting from the measured site [@Thiel2017; @Friedman2017]. We conclude our study by generalising our spectral results to this type of open quantum systems described in general by the Lindblad master equation, providing a natural link with the recent extension of the spectral theory of metastability to quantum systems [@Macieszczak2016]. We illustrate this generalization by computing numerically the stationary state of a model of coherent hopping in one dimension realizing the reset quantum 