具有相互作用珠子的项链：等周问题
Necklaces with interacting beads: isoperimetric problems
摘要：
我们讨论一对等周问题，乍一看似乎是无关的。第一个是古典的：将$N$相同的点电荷放置在相同弧长距离处的闭合曲线$Γ$上，并询问能量最小值，即，如果单独留下，环采取何种形状。第二个问题来自量子力学：我们在$L^2(\mathbb{R}^d), d=2,3,$中取一个薛定谔算符，其中$N$同点相互作用以所述方式放置在一个环中，并询问\emph{maximizes}的基态能量的配置。我们减少他们两个几何不等式，涉及弦的$Γ$，它将表明，尖锐的局部极值是在这两种情况下达到的$Γ$的形式，一个定期（平面）多边形，这样的$Γ$解决这两个问题也是全球性的。
英文摘要：
We discuss a pair of isoperimetric problems which at a glance seem to be unrelated. The first one is classical: one places $N$ identical point charges at a closed curve $Γ$ at the same arc-length distances and asks about the energy minimum, i.e. which shape does the loop take if left by itself. The second problem comes from quantum mechanics: we take a Schrödinger operator in $L^2(\mathbb{R}^d), d=2,3,$ with $N$ identical point interaction placed at a loop in the described way, and ask about the configuration which \emph{maximizes} the ground state energy. We reduce both of them to geometric inequalities which involve chords of $Γ$; it will be shown that a sharp local extremum is in both cases reached by $Γ$ in the form of a regular (planar) polygon and that such a $Γ$ solves the two problems also globally.
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