Perturbed Bessel operators

TL;DR

This paper analyzes perturbed Bessel operators with complex potentials, constructing solutions with specific behaviors, and classifies all their closed realizations as holomorphic families of operators.

Contribution

It provides a comprehensive analysis of perturbed Bessel operators with complex potentials, including solution construction and classification of all closed realizations.

Findings

Constructed solutions with prescribed behaviors near zero.

Classified all closed realizations of the operators.

Established holomorphic dependence of realizations on parameters.

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.