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On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to fermion nodes. (English) Zbl 1209.65010

The author considers Schrödinger operators and presents a probabilistic interpretation of the variation of the Dirichlet groundstate energy when the associated domain is perturbed. This interpretation relies on the distribution on the boundary of a stopped random process with Feynman-Kac weights. Practical computations require in addition the explicit approximation of the normal derivative of the groundstate on the boundary. This formulation is used in the case of the so-called fixed node approximation of Fermion groundstates, defined by the bottom eigenelements of the Schrödinger operator of a Fermionic system with Dirichlet conditions on the nodes of an initially guessed skew-symmetric function. It is shown that shape derivatives of the fixed node energy vanishes if and only if either (i) the distribution on the nodes of the stopped random process is symmetric; or (ii) the nodes are exactly the zeros of a skew-symmetric eigenfunction of the operator. A consistent approximation of the shape derivative of the fixed node energy that can be computed with a Monte-Carlo algorithm is presented.

MSC:

65C35 Stochastic particle methods
60H30 Applications of stochastic analysis (to PDEs, etc.)
65C05 Monte Carlo methods
35J10 Schrödinger operator, Schrödinger equation
35P25 Scattering theory for PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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References:

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