Square Laplacian perturbed by inverse fourth-power potential II. Holomorphic family of type (A) (complex case)

Abstract

It is proved that $\{{\mathrm{\Delta }}^2+\kappa |x{|}^{-4};\kappa \in \sum _{}^{\mathrm{c}}\}$ in $L^2({\mathbf{ℝ}}^N)$ forms a holo-morphic family of type (A), where $\sum$ is a closed and convex subset of $ℂ$. In particular, the $m$-accretivity of ${\mathrm{\Delta }}^2+\kappa |x{|}^{-4}$ in $L^2({\mathbf{ℝ}}^N)$ is established as an application of the perturbation theorem for linear $m$-accretive operators. The key lies in two inequalities derived by positive semi-definiteness of Gram matrix.