Higher order logarithms of Bessel operators and an extension problem

Abstract

We consider the Bessel operator defined by \[ B_\lambda =-\frac{d^2}{dx^2}+\frac{\lambda^2-1/4}{x^2}, \] on $(0,\infty)$, with $\lambda>-1$. We study the fractional power $B_\lambda^s$, $s\in (-1,1)$, $s\neq 0$, and the logarithm $\log^kB_\lambda $, $k\in \mathbb N$, of $B_\lambda$. We obtain pointwise representations of these operators and asymptotic Taylor expansions of the operators $B_\lambda^s$ in terms of logarithmic operators $\log^kB_\lambda $. We also obtain $\log B_\lambda$ as the solution of an extension problem.