  The Diffusion Monte Carlo method with constant number of walkers, also called
Stochastic Reconfiguration as well as Sequential Monte Carlo, is a widely used
Monte Carlo methodology for computing the ground-state energy and wave function
of quantum systems. In this study, we present the first mathematically rigorous
analysis of this class of stochastic methods on non necessarily compact state
spaces, including linear diffusions evolving in quadratic absorbing potentials,
yielding what seems to be the first result of this type for this class of
models. We present a novel and general mathematical framework with easily
checked Lyapunov stability conditions that ensure the uniform-in-time
convergence of Diffusion Monte Carlo estimates towards the top of the spectrum
of Schr\"odinger operators. For transient free evolutions, we also present a
divergence blow up of the estimates w.r.t. the time horizon even when the
asymptotic fluctuation variances are uniformly bounded. We also illustrate the
impact of these results in the context of generalized coupled quantum harmonic
oscillators with non necessarily reversible nor stable diffusive particle and a
quadratic energy absorbing well associated with a semi-definite positive matrix
force.
