Regularized traces for a class of singular operators (Q1603158)

The author considers the operator \((-1)^n \frac{d^{2n}}{dx^{2n}} +x\) in \(L^2[0,\infty)\), with the boundary conditions \(y^{(k_m)}(0)=0\), \(k_n<k_{n-1} <\dots <k_1<2n.\) Let \(\{\lambda_k\}\) be the eigenvalues. The complete asymptotic expansion of \(\lambda_k^{-\sigma}\) as \(k\to \infty\) is found for any complex number \(\sigma\).