On the classical formula for the first regularized trace of the Laplace operator with an odd potential on the sphere (Q1280066)
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scientific article; zbMATH DE number 1259424
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the classical formula for the first regularized trace of the Laplace operator with an odd potential on the sphere |
scientific article; zbMATH DE number 1259424 |
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On the classical formula for the first regularized trace of the Laplace operator with an odd potential on the sphere (English)
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15 March 1999
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\textit{V. Guillemin} [Adv. Math. 27, 273-386 (1978; Zbl 0433.35052)] has investigated the eigenvalues of \(B+P\) where \(B\) is the Laplace-Beltrami operator on \(S^{N-1}\) and \(P\) denotes multiplication by a smooth odd function. One consequence of this work in the special case \(N=3\) yields the existence of \(2n+1\) eigenvalues \(\mu_{n,i}\) \((i=0, \dots,2n)\) near \(n(n+1)\) such that \[ \sum^{2n}_{i=0}\mu_{n,i} =(2n+1) n(n+1) +O(n^{-1}). \] The authors improve the remainder term to be \(O(n^{-3/2}\ln n)\) thus allowing to compute the trace mentioned in the title.
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0.8905506730079651
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0.8889729976654053
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