Spectrum asymptotics of a perturbed Laplace-Beltrami operator on a three-dimensional sphere
This paper studies the asymptotics of the eigenvalues of the operator \(H=H_0 +V\) in \(L^2(S^3)\), where \(S^3\) is the unit three-dimensional sphere. Here, \(H_0\) is the Laplace-Beltrami operator and \(V\) is the operator of multiplication by a measurable real-valued function \(V(\omega)\in C(S^3)\). The spectrum of the operator \(H_0\) is discrete and consists of the eigenvalues \(\lambda_n = n(n + 2)\) of multiplicity \((n + 1)^2\). The main result of the paper is an asymptotic formula for \( \sum\limits_{k=1}^{(n+1)^2} \mu_n^{(k)}\), where \(\{ \mu_n^{(k)} \}_{k=1}^{(n+1)^2}\) is the set of eigenvalues of \(H\) contained in the interval \([\lambda_n - V_0, \lambda_n + V_0]\), where \(V_0=\max\limits_{\omega\in S^3}|V(\omega)|\). The proof is based on the investigation of the properties of the unperturbed resolvent \((H_0-z)^{-1}\).
scientific article; zbMATH DE number 6032766