We derive and numerically implement various asymptotic approximations for the lowest or principal eigenvalue of the Laplacian with a periodic arrangement of localised traps of small \[\mathcal{O}(\varepsilon )\] spatial extent that are centred at the lattice points of an arbitrary Bravais lattice in \[{\mathbb{R}^2}\]. The expansion of this principal eigenvalue proceeds in powers of \[\nu \equiv - 1/\log (\varepsilon {d_c})\], where dc is the logarithmic capacitance of the trap set. An explicit three-term approximation for this principal eigenvalue is derived using strong localised perturbation theory, with the coefficients in this series evaluated numerically by using an explicit formula for the source-neutral periodic Green’s function and its regular part. Moreover, a transcendental equation for an improved approximation to the principal eigenvalue, which effectively sums all the logarithmic terms in powers of v, is derived in terms of the regular part of the periodic Helmholtz Green’s function. By using an Ewald summation technique to first obtain a rapidly converging infinite series representation for this regular part, a simple Newton iteration scheme on the transcendental equation is implemented to numerically evaluate the improved ‘log-summed’ approximation to the principal eigenvalue. From a numerical computation of the PDE eigenvalue problem defined on the fundamental Wigner–Seitz (WS) cell for the lattice, it is shown that the three-term asymptotic approximation for the principal eigenvalue agrees well with the numerical result only for a rather small trap radius. In contrast, the log-summed asymptotic result provides a very close approximation to the principal eigenvalue even when the trap radius is only moderately small. For a circular trap, the first few transcendental correction terms that further improves the log-summed approximation for the principal eigenvalue are derived. Finally, it is shown numerically that, amongst all Bravais lattices with a fixed area of the primitive cell, the principal eigenvalue is maximised for a regular hexagonal arrangement of traps.

我们推导并数值实现了拉普拉斯算子的最低或主特征值的各种渐近近似，其中小\[\mathcal{O}(\varepsilon )\]空间范围的局部陷阱的周期性排列以\[{\mathbb{R}^2}\]中的任意布拉维格。此主特征值的展开以\[\nu \equiv - 1/\log (\varepsilon {d_c})\]的幂进行，其中d c是陷阱集的对数电容。使用强局部微扰理论推导出该主特征值的显式三项近似，该系列中的系数通过使用源中性周期格林函数及其规则部分的显式公式进行数值评估。此外，用于改进近似主特征值的超越方程，它有效地将所有对数项以v的幂求和, 是根据周期性亥姆霍兹格林函数的规则部分导出的。通过使用 Ewald 求和技术首先获得该规则部分的快速收敛无限级数表示，实现了超越方程的简单牛顿迭代方案，以数值评估改进的对主要特征值的“对数求和”近似。通过对晶格的基本 Wigner-Seitz (WS) 单元上定义的 PDE 特征值问题的数值计算，表明主特征值的三项渐近逼近仅与相当小的陷阱的数值结果一致半径。相反，即使陷阱半径只是适度小，对数和渐近结果也提供了对主特征值的非常接近的近似值。对于圆形陷阱，导出了进一步改进主要特征值的对数和近似的前几个超越校正项。最后，数值表明，在所有具有固定原始单元面积的 Bravais 晶格中，对于规则的六边形排列的陷阱，主特征值最大化。