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On the control of an interacting particle estimation of Schrödinger ground states. (English) Zbl 1174.60045

The paper concerns 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 \(L(h)+Vh=\lambda h\). 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 a non-asymptotic control of the \(\mathbb{L}^p\) (resp., the bias) deviations of this estimator is obtained with the rate of convergence in \(1/\sqrt{N}\) (resp., \(1/N\)).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
35Q40 PDEs in connection with quantum mechanics
60J35 Transition functions, generators and resolvents
65C35 Stochastic particle methods
81-08 Computational methods for problems pertaining to quantum theory
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