Complex powers of vector valued operators and their application to asymptotic behavior of eigenvalues (Q911005)

From the authors introduction: The asymptotic behaviour of the counting function N(\(\lambda)\) of eigenvalues for a class of differential operators containing the operator \(A=-\Delta +(| x|^ 2+1)^ py^ 2,\) \((x,y)\in {\mathbb{R}}^ p\times {\mathbb{R}}\), is considered. For this type of operators, it is known that the first term of N(\(\lambda)\) as \(\lambda\) \(\to \infty\) is closely related to the first singularity of the meromorphic extension \(Z_ A(s)\) of the trace Tr \(A^{-s}\). It is also shown that the second term of N(\(\lambda)\) can be found by using the extended Ikehara Tauberian theorem.