We obtain new Faber–Krahn-type inequalities for certain perturbations of the Dirichlet Laplacian on a bounded domain. First, we establish a two- and three-dimensional Faber–Krahn inequality for the Schrödinger operator with point interaction: the optimizer is the ball with the point interaction supported at its center. Next, we establish three-dimensional Faber–Krahn inequalities for a one- and two-body Schrödinger operator with attractive Coulomb interactions, the optimizer being given in terms of Coulomb attraction at the center of the ball. The proofs of such results are based on symmetric decreasing rearrangement and Steiner rearrangement techniques; in the first model, a careful analysis of certain monotonicity properties of the lowest eigenvalue is also needed.

中文翻译：

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带点和库仑相互作用的薛定ding算子的Faber-Krahn不等式

对于有界域上的Dirichlet Laplacian的某些摄动，我们获得了新的Faber-Krahn型不等式。首先，我们为具有点交互作用的Schrödinger算子建立了二维和三维Faber-Krahn不等式：优化器是在中心支持点交互作用的球。接下来，我们为具有吸引力库仑相互作用的一体和两体Schrödinger算子建立三维Faber-Krahn不等式，并根据球中心的库仑引力给出优化器。这种结果的证明是基于对称递减重排和Steiner重排技术。在第一个模型中，还需要仔细分析最低特征值的某些单调性。