On the discrete spectrum of periodic differential operators with a distant perturbation (Q352187)

The author considers eigenvalues of a perturbed self-adjoint periodic differential operator of even order in \(L^2({\mathbb R}^d;{\mathbb C}^n)\). The perturbations are abstract symmetric operators. The case is considered when the perturbing operators are ``localized'' in domains that are far away from each other. The author announces the following results:{\parindent=0.5cm\begin{itemize}\item[-] the essential spectrum is stable under perturbations; \item[-] complete asymptotic expansions of the eigenvalues and the corresponding eigenfunctions of the perturbed operator; \item[-] uniform convergence of these asymptotic series and explicit formulas for their coefficients. \end{itemize}}