有限点相互作用族的一个最优化问题
An optimization problem for finite point interaction families
摘要：
我们考虑了一个家庭的$N$点相互作用的相同强度的限制到一个流形的旋转对称，一个圆圈或一个球的频谱问题，并要求配置，优化相应的奇异薛定谔算子的基态能量。在圆的情况下，主特征值急剧最大化，如果点的相互作用分布在相等的距离。类似的问题，为领域是困难得多，并减少到修改汤姆逊问题，我们已经能够表明独特的最大化配置$N=2,\,3,\,4,\,6,\,12$。我们还讨论了具有周期边界条件的一维点相互作用的优化问题。我们证明了当相互作用是吸引的时，等距离分布会产生最大基态本征值，而在排斥的情况下，对于弱耦合和强耦合我们得到了同样的结果，并且我们推测这是普遍有效的。
英文摘要：
We consider the spectral problem for a family of $N$ point interactions of the same strength confined to a manifold with a rotational symmetry, a circle or a sphere, and ask for configurations that optimize the ground state energy of the corresponding singular Schrödinger operator. In case of the circle the principal eigenvalue is sharply maximized if the point interactions are distributed at equal distances. The analogous question for the sphere is much harder and reduces to a modification of Thomson problem; we have been able to indicate the unique maximizer configurations for $N=2,\,3,\,4,\,6,\,12$. We also discuss the optimization for one-dimensional point interactions on an interval with periodic boundary conditions. We show that the equidistant distributions give rise to maximum ground state eigenvalue if the interactions are attractive, in the repulsive case we get the same result for weak and strong coupling and we conjecture that it is valid generally.
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