The eigenvalue problem for the negative Laplace operator in two dimensions is classical in mathematics and physics. Nevertheless, analytical methods for estimating the eigenvalues are still of much current interest. In this work, a modified perturbation method is formulated by applying perturbation method, reflection method, and the Fredholm alternative theorem. The method provides the asymptotic expansion formulas of the lowest eigenvalue to bounded doubly connected regions having the inner boundary which encloses a region with the maximum dimension of 2c, c 1. The first three order terms of the asymptotic expansion formulas are found explicitly by correcting the inner and outer boundary conditions alternatively and by applying the generalized Green's functions. The relations between the first three order terms of the asymptotic expansion formulas and geometric properties of the regions are also investigated.
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机译：负Laplace算子在二维上的特征值问题在数学和物理学上都是经典的。尽管如此，用于估计特征值的分析方法仍然是当前的热点。在这项工作中，通过应用摄动方法，反射方法和Fredholm替代定理，提出了一种改进的摄动方法。该方法为具有内边界的有界双重连接区域提供了最低特征值的渐近展开公式，该内边界包围了最大尺寸为2 c ， c 展开▼
