Some spectral properties of the perturbed polyharmomic operator

Abstract:

We deal with the polyharmonic operator perturbed by a potential, decreasing at infinity as ${\left | x \right |^{ - \sigma }}$. Under some conditions we obtain the absence of eigenvalues in a neighbourhood of the point $z = 0$, the existence of the strong limit and the asymptotic expansion of the corresponding resolvent ${R_z}$, considered in weighted ${L^2}$-spaces, as $z \to 0$, where $z$ is the spectral parameter.