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

We consider a general Schrödinger operator $L+V$ on a domain E⊂ Rᵈ and its associated positive ground state h solution to the maximal eigenvalue problem L(h) +Vh=λ h. 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= Rᵈ), we get a nonasymptotic control of the Lᵖ 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.