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Lie group methods for eigenvalue function. (English) Zbl 1456.58023

Summary: By considering a \(C^\infty\) structure on the ordered non-increasing of elements of \(\mathbb R^n\), we show that it is a differentiable manifold. By using of Lie groups, we show that eigenvalue function is a submersion. This fact is used to prove some results. These results is applied to prove a few facts about spectral manifolds and spectral functions. Orthogonal matrices act on the real symmetric matrices as a Lie transformation group. This fact, also, is used to prove the results.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58A05 Differentiable manifolds, foundations
22E20 General properties and structure of other Lie groups
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Full Text: Euclid