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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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