On A Family of Isospectral Pure-Birth Processes

The investigation of the spectra of finite Markov generators was first motivated by the quest of quantitative bounds on the convergence to equilibrium of the corresponding processes, see for instance the reference book Levin et al. (2009). There is also a classification reason: what are the possible spectra of Markov generators?, and for such a given spectrum, is there a simple representative Markov process? This structural question is related to the previous motivation, as ergodic finite isospectral Markov generators can be intertwined and under certain circumstances this relation enables good transfers of information about the speed of convergence, see e.g. Miclo (2018); Miclo and Patie (2021). Our goal here goes in this general direction, by providing a simple family of Markov generators for real spectra with geometric multiplicity 1.