Perturbed Bessel operators

Abstract

We study perturbed Bessel operators $L_{m^2}=- \partial^2_x + ( m^2 - \frac14 )\frac{1}{x^2} + Q(x)$ on $L^2]0,\infty[$, where $m\in\mathbb{C}$ and $Q$ is a complex locally integrable potential. Assuming that $Q$ is integrable near $\infty$ and $x\mapsto x^{1-\varepsilon}Q(x)$ is integrable near $0$, with $\varepsilon\ge0$, we construct solutions to $L_{m^2} f = - k^2 f$ 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 $L_{m^2}$ and show that they can be organized as holomorphic families of closed operators.