Rudnick Zeév, Wigman Igor, Yesha Nadav
School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel.
Department of Mathematics, King's College London, London, UK.
Commun Math Phys. 2021;388(3):1603-1635. doi: 10.1007/s00220-021-04248-y. Epub 2021 Nov 2.
Let be a bounded planar domain, with piecewise smooth boundary . For , we consider the Robin boundary value problem where is the derivative in the direction of the outward pointing normal to . Let be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps For a wide class of planar domains we show that there is a limiting mean value, equal to and in the smooth case, give an upper bound of and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.
设(\Omega)是一个有界平面区域，其边界(\partial\Omega)是分段光滑的。对于(\alpha\in\mathbb{R})，我们考虑罗宾边值问题(\begin{cases}-\Delta u=\lambda u&\text{在 }\Omega 内\\frac{\partial u}{\partial\nu}+\alpha u = 0&\text{在 }\partial\Omega 上\end{cases})，其中(\frac{\partial u}{\partial\nu})是(u)沿指向(\Omega)外部的法向方向的导数。设(\lambda_n)是相应的特征值。本文的目的是研究罗宾 - 诺伊曼间隙(\lambda_{n + 1}-\lambda_n)。对于一大类平面区域，我们证明存在一个极限平均值，等于(\frac{4\pi}{|\partial\Omega|})，并且在光滑情况下，给出(\lambda_{n + 1}-\lambda_n)的一个上界和一个一致下界。对于遍历台球问题，我们证明沿着一个密度为(1)的子序列，间隙收敛到平均值。我们还得到了矩形（在矩形中我们有一个一致上界）和圆盘（在圆盘中我们改进了一般上界）的进一步性质。
