An extension of the Ikehara Tauberian theorem and its application
In this article [which is the fifth of a series devoted by the author to this thema, see e.g. J. Funct. Anal. 87, No. 2, 294-320 (1989; Zbl 0697.47011)] the author presents an extension of the Ikehara's Tauberian theorem relating to a selfadjoint operator with compact resolvent of the properties of the meromorphic function \(s\mapsto\text{Tr } P^{-s}\) to the asymptotic behavior, when \(\lambda\) is large, of the counting function \(N_P(\lambda)\) of the eigenvalues of \(P\) which are less than \(\lambda\). This is applied to the case when \(P\) takes the form \[ P=-\Delta_x-\Delta_y+ (1+|x|^2)^p|y|^{2q}, \] on \(\mathbb{R}^n_x\times \mathbb{R}^m_y\). This example was previously analyzed by \textit{D. Robert} [J. Math. Pures Appl. 61, 275-300 (1982; Zbl 0511.35069)] but without the same control on the remainder term.
scientific article; zbMATH DE number 897186