Perturbed Bessel operators. Boundary conditions and closed realizations

Abstract

We study perturbed Bessel operators Lm2=−∂x2+(m2−14)1x2+Q(x) on L2]0,∞[, where m∈C and Q is a complex locally integrable potential. Assuming that Q is integrable near ∞ and x↦x1−εQ(x) is integrable near 0, with ε≥0, we construct solutions to Lm2f=−k2f with prescribed behaviors near 0. The special cases m=0 and k=0 are included in our analysis. Our proof relies on mapping properties of various Green's operators of the unperturbed Bessel operator. Then we determine all closed realizations of Lm2 and show that they can be organized as holomorphic families of closed operators.