correspondence that exists between resets and renewals to obtain renewal representations of both time-dependent and stationary distributions, in addition to first-passage statistics. Modified Fokker–Planck and Feynman–Kac equations with additional source and sink terms describing the evolution of these distributions and statistics have also been obtained (see, e.g., [@evans2011b; @dharmaraja2015; @meylahn2015b]) and can be solved explicitly for some simple models, including reset versions of Brownian motion [@evans2011] and the Ornstein-Uhlenbeck process [@pal2015].

In this paper, we present a different approach to reset processes based on the spectral properties of their generator or master operator. Our main result is a relation between the spectrum of the generator of a reset process and that of its reset-free counterpart. More precisely, we show that the real part of the eigenvalues of the generator are shifted for all non-stationary modes by the reset rate, while the corresponding eigenstates, representing the dynamical modes, are not modified. We also provide explicit expressions for the stationary state and current of reset processes involving weighted sums over the dynamical modes, which are applied to two prototypical models, namely, the totally asymmetric random walk in one dimension, related to queuing, and the one-dimensional Brownian motion. The results obtained clearly explain 