On the Control of an Interacting Particle Estimation of Schrödinger Ground States

We consider a general Schrödinger operator L + V on a domain E ⊂ R, and its associated positive ground state h solution to the maximal eigenvalue problem L(h) + V h = λh. In this work, an interacting particle model approximating the pair (h, λ) is studied. When V ≤ 0, a basic version of this particle system consists of N walkers evolving independently according to the Markov generator L, each walker dying at a rate given by the value of the potential |V | at the walker’s current location; when a walker dies, any other one splits in two. The long time distribution of the particle system is then an estimator of h. Under some reasonable assumptions (with examples for E = R), we get a non-asymptotic control of the L deviations (resp. the bias) of this estimator with the genuine rate of convergence in 1/ √ N (resp.1/N). We also compute explicitly the asymptotic standard deviation of the estimation of λ, which remains bounded in usual mild situations.