中文摘要：

本征值问题是自然科学中基本运算之一，对于超大矩阵的对角化是当今许多科学问题的瓶颈。在应用原子核壳模型理论研究较重的原子核结构时，因为壳模型组态太大，通常的方法是基于各种物理考虑做某些组态截断，另一个思路是利用新的算法和飞速发展的计算机资源对这些大矩阵对角化或者近似对角化。总结了本课题组近年来在壳模型哈密顿量本征值近似方面研究的主要结果，包括最低本征值半经验公式及多种外推方法、本征值与对角元的相关性等。

英文摘要：

The eigenvalue problem is one of the fundamental issues of sciences. Many research fields have been challenged by diagonalizing huge matrices. The nuclear structure theorists face this problem in studies of medium-heavy nuclei in terms of the nuclear shell model, in which the configuration space is too gigantic to handle. Thus one usually truncates the nuclear shell model configuration space based on various considerations. Another approach is to make use of super computers by various algorithms, and/or to obtain approximate eigenvalues. In this paper we review our recent efforts in obtaining approximate eigenvalues of the nuclear shell model Hamiltonian, with the focus on our semi-empirical approach and a number of extrapolation approaches towards predicting the lowest eigenvalue, as well as strong correlation between the sorted eigenvalues and the diagonal matrix elements, and so on.