The non-Hermitian Bethe-Salpeter eigenvalue problem is a structured eigenproblem, with real eigenvalues coming in pairs $\{\lambda,-\lambda\}$ where the corresponding pair of eigenvectors are closely related, and furthermore the left eigenvectors can be trivially obtained from the right ones. We exploit these properties to devise three variants of structure-preserving Lanczos eigensolvers to compute a subset of eigenvalues (those of either smallest or largest magnitude) together with their corresponding right and left eigenvectors. For this to be effective in real applications, we need to incorporate a thick-restart technique in a way that the overall computation preserves the problem structure. The new methods are validated in an implementation within the SLEPc library using several test matrices, some of them coming from the Yambo materials science code.

翻译：非厄米Bethe-Salpeter特征值问题是一种结构化特征问题，其特征值为实数且成对出现$\{\lambda,-\lambda\}$，其中对应的特征向量对紧密相关，且左特征向量可直接由右特征向量简单导出。我们利用这些性质，设计了三种保持结构的Lanczos特征求解器变体，用于计算部分特征值（最小或最大模值）及其对应的右、左特征向量。为使该方法在实际应用中有效，我们需以保持问题结构的方式引入厚重启技术。新方法在SLEPc库中的实现通过了多个测试矩阵的验证，其中部分测试矩阵来源于Yambo材料科学代码。