The Diffusion quantum Monte Carlo method with constant number of walkers, also called Stochastic Reconfiguration as well as Sequential Monte Carlo, is a widely used genetic type Monte Carlo methodology for computing the ground-state energy and wave function of quantum systems. In this study, we present a general probabilistic framework with easily checked regularity conditions that ensure the uniform-in-time convergence of Diffusion Monte Carlo estimates towards the top of the spectrum of Schrödinger operators. We develop a novel powerful Lyapunov approach that applies to non necessarily compact state spaces, including linear diffusions evolving in quadratic absorbing potentials. We also illustrate the impact of these results in the context of coupled quantum harmonic oscillators, yielding what seems to be the first result of this type for this class of models.