Abstract The eigenvalues of the Laplace-Beltrami operator on the torus embedded in three-dimensional euclidean space are investigated. These eigenvalues are determined by the eigenvalues of Sturm-Liouville problems with separated boundary conditions. Eigenvalues are approximated by those of generalized eigenvalue problems involving tridiagonal matrices. A non-coexistence result is proved. The behavior of eigenvalues is studied when the ratio of inner and outer radius of the torus approaches zero or one.

中文翻译：

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嵌入式环面上的拉普拉斯-贝尔特拉米算子

摘要 研究了嵌入三维欧几里得空间的圆环上的拉普拉斯-贝尔特拉米算子的特征值。这些特征值由具有分离边界条件的 Sturm-Liouville 问题的特征值确定。特征值由涉及三对角矩阵的广义特征值问题的特征值逼近。证明了不共存的结果。当环的内外半径比接近零或一时，特征值的行为被研究。