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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.01620 |
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| _version_ | 1866909268485079040 |
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| author | Volkmer, Hans |
| author_facet | Volkmer, Hans |
| contents | The Laplace-Beltrami operator on (the surface of) a triaxial ellipsoid admits a sequence of real eigenvalues diverging to plus infinity. By introducing ellipsoidal coordinates, this eigenvalue problem for a partial differential operator is reduced to a two-parameter regular Sturm-Liouville problem involving ordinary differential operators. This two-parameter eigenvalue problem has two families of eigencurves whose intersection points determine the eigenvalues of the Laplace-Beltrami operator. Eigenvalues are approximated numerically through eigenvalues of generalized matrix eigenvalue problems. Ellipsoids close to spheres are studied employing Lamé polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01620 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Laplace-Beltrami Operator on the Surface of the Ellipsoid Volkmer, Hans Classical Analysis and ODEs 34B30, 34L15 The Laplace-Beltrami operator on (the surface of) a triaxial ellipsoid admits a sequence of real eigenvalues diverging to plus infinity. By introducing ellipsoidal coordinates, this eigenvalue problem for a partial differential operator is reduced to a two-parameter regular Sturm-Liouville problem involving ordinary differential operators. This two-parameter eigenvalue problem has two families of eigencurves whose intersection points determine the eigenvalues of the Laplace-Beltrami operator. Eigenvalues are approximated numerically through eigenvalues of generalized matrix eigenvalue problems. Ellipsoids close to spheres are studied employing Lamé polynomials. |
| title | The Laplace-Beltrami Operator on the Surface of the Ellipsoid |
| topic | Classical Analysis and ODEs 34B30, 34L15 |
| url | https://arxiv.org/abs/2312.01620 |