On Eigenvalue Generic Properties of the Laplace-Neumann Operator
| Autor: | Marcus A. M. Marrocos, José N. V. Gomes |
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| Rok vydání: | 2015 |
| Předmět: |
Mathematics - Differential Geometry
010102 general mathematics Position operator Spectrum (functional analysis) Mathematical analysis General Physics and Astronomy Riemannian manifold Mathematics::Spectral Theory Compact operator 01 natural sciences Semi-elliptic operator Operator (computer programming) Laplace–Beltrami operator Differential Geometry (math.DG) Spectrum of a matrix 0103 physical sciences FOS: Mathematics 010307 mathematical physics Geometry and Topology 0101 mathematics Mathematical Physics Mathematics |
| DOI: | 10.48550/arxiv.1510.07067 |
| Popis: | We establish the existence of analytic curves of eigenvalues for the Laplace-Neumann operator through an analytic variation of the metric of a compact Riemannian manifold $M$ with boundary by means of a new approach rather than Kato's method for unbounded operators. We obtain an expression for the derivative of the curve of eigenvalues, which is used as a device to prove that the eigenvalues of the Laplace-Neumann operator are generically simple in the space $\mathcal{M}^k$ of all $C^k$ Riemannian metrics on $M$. This implies the existence of a residual set of metrics in $\mathcal{M}^k$, which make the spectrum of the Laplace-Neumann operator simple. We also give a precise information about the complementary of this residual set, as well as about the structure of the set of the deformation of a Riemannian metric which preserves double eigenvalues. Comment: Final version which has been accepted for publication in Journal of Geometry and Physics |
| Databáze: | OpenAIRE |
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