Asymptotic behavior of eigenvalues for a class of pseudodifferential operators on \({\mathbb{R}{}}^ n\) (Q1209674)

The author considers a class of pseudodifferential operators \(P\) whose symbol has an asymptotic expansion by quasi-homogeneous symbols and whose principal symbol is degenerate on a submanifold. This class contains in particular Schrödinger operators with electric potential and magnetic fields. Under appropriate conditions the author gets the asymptotic behavior of the counting function of the eigenvalues with a Weyl term and some control of the remainder term. The method is based on the technique of complex powers of the operator initiated by \textit{R. T. Seeley} [Proc. Symp. Pure Math. 10, 288-307 (1967; Zbl 0159.155)].