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

Abstract

We consider a general Schrödinger operator $L+V$ on a domain $E\subset {\mathbb R}^{d}$ and its associated positive ground state h solution to the maximal eigenvalue problem $L(h) +Vh=\lambda h$. In this work, an interacting particle model approximating the pair $(h,\lambda)$ is studied. When $V \leq 0$, 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 $E={\mathbb R}^{d}$), we get a nonasymptotic control of the $\mathbb{L}^{p}$ deviations (resp., the bias) of this estimator with the genuine rate of convergence in $1/\sqrt{N}$ (resp., $1/\sqrt{N}$). We also compute explicitly the asymptotic standard deviation of the estimation of λ, which remains bounded in usual mild situations.