On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes.

摘要: This paper considers Schrodinger operators, and presents a probabilistic interpretation of the variation (or shape derivative) Dirichlet groundstate energy when associated domain is perturbed. relies on distribution boundary stopped random process with Feynman-Kac weights. Practical computations require in addition explicit approximation normal derivative boundary. We then propose to use this formulation case so-called fixed node Fermion groundstates, defined by bottom eigenelements operator Fermionic system conditions nodes (the set zeros) an initially guessed skew-symmetric function. show that derivatives vanishes if only either (i) symmetric; or (ii) are exactly zeros eigenfunction operator. can be computed Monte-Carlo algorithm, which referred as Nodal (NMC). The latter also holds.