开弧上奇异相互作用的谱等周不等式
Spectral isoperimetric inequalities for singular interactions on open arcs
摘要：
考虑了具有吸引$δ$相互作用的二维薛定谔算子的最低本征值的几何优化问题.在弧长固定的约束下，证明了极大元是一条线段，相应的谱等周不等式是严格的。我们还表明，在最优化问题相同的光谱量，但与固定端点的约束，优化是连接它们的线段。作为结果的$δ$ -相互作用，我们得到的线段也是最大值的最优化问题的最低特征值的罗宾拉普拉斯在一个平面上的狭缝沿一个开放的弧的固定长度。
英文摘要：
We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive $δ$-interaction supported on an open arc with two free endpoints. Under a constraint of fixed length of the arc, we prove that the maximizer is a line segment, the respective spectral isoperimetric inequality being strict. We also show that in the optimization problem for the same spectral quantity, but with the constraint of fixed endpoints, the optimizer is the line segment connecting them. As a consequence of the result for $δ$-interaction, we obtain that a line segment is also the maximizer in the optimization problem for the lowest eigenvalue of the Robin Laplacian on a plane with a slit along an open arc of fixed length.
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