In the setting of the fractional quantum Hall effect we study the effects of strong, repulsive two-body interaction potentials of short range. We prove that Haldane’s pseudo-potential operators, including their pre-factors, emerge as mathematically rigorous limits of such interactions when the range of the potential tends to zero while its strength tends to infinity. In a common approach the interaction potential is expanded in angular momentum eigenstates in the lowest Landau level, which amounts to taking the pre-factors to be the moments of the potential. Such a procedure is not appropriate for very strong interactions, however, in particular not in the case of hard spheres. We derive the formulas valid in the short-range case, which involve the scattering lengths of the interaction potential in different angular momentum channels rather than its moments. Our results hold for bosons and fermions alike and generalize previous results in [ 6 ], which apply to bosons in the lowest angular momentum channel. Our main theorem asserts the convergence in a norm-resolvent sense of the Hamiltonian on the whole Hilbert space, after appropriate energy scalings, to Hamiltonians with contact interactions in the lowest Landau level.

中文翻译：

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Haldane 伪电势在短程相互作用系统中的出现

在分数量子霍尔效应的背景下，我们研究了短程强排斥性二体相互作用势的影响。我们证明，当势的范围趋于零而其强度趋于无穷大时，霍尔丹的伪势算子，包括它们的前因数，作为这种相互作用的数学严格限制出现。在一种常见的方法中，相互作用势在最低朗道能级的角动量本征态中扩展，这相当于将前因数作为势的矩。然而，这种程序不适用于非常强的相互作用，尤其不适用于硬球。我们推导出在短程情况下有效的公式，这涉及相互作用势在不同角动量通道中的散射长度而不是其矩。我们的结果同样适用于玻色子和费米子，并概括了 [6] 中先前的结果，这些结果适用于最低角动量通道中的玻色子。我们的主要定理断言了整个希尔伯特空间上哈密顿量在范数解析意义上的收敛，经过适当的能量缩放，到在最低朗道能级上具有接触相互作用的哈密顿量。