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A law of large numbers and a central limit theorem for the Schrödinger operator with zero-range potentials

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We consider the Schrödinger operator with zero-range potentials onN points of three-dimensional space, independently chosen according to a common distributionV(x). Under some assumptions we prove that, whenN goes to infinity, the sequence converges to a Schrödinger operator with an effective potential. The fluctuations around the limit operator are explicitly characterized.

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Figari, R., Holden, H. & Tetab, A. A law of large numbers and a central limit theorem for the Schrödinger operator with zero-range potentials. J Stat Phys 51, 205–214 (1988).

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