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The formula of the regularized trace for the Laplace-Beltrami operator with odd potential on the sphere \(S^ 2\). (English. Russian original) Zbl 0834.35093
Math. Notes 56, No. 1, 699-703 (1994); translation from Mat. Zametki 56, No. 1, 71-77 (1994).
The main result of the present paper is the following theorem: Let \(\{\lambda_k \}^\infty_{k= 0}\) be eigenvalues of the Laplace- Beltrami operator \(\Delta\) on the sphere \(S^2\), indexed taking account of their multiplicity, \(\lambda_k\leq \lambda_{k+1}\), and let \(\{\mu_k \}^\infty_{k=0}\) be eigenvalues of the operator \(-\Delta +q\), where \(q\) is the operator of multiplication by a smooth odd function given on the sphere \(S^2\). Then the series with the common term \((\lambda_k- \mu_k)\) converges and its sum is equal to \[ \sum_{k=0}^\infty (\mu_k- \lambda_k)=- {1\over {8\pi}} \int_{S^2} q^2 dS. \]

MSC:
35P05 General topics in linear spectral theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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References:
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