Fredholm properties of radially symmetric, second order differential operators (Q646385)

The authors analyze the Fredholm properties of radially symmetric elliptic operators \[ \begin{aligned} \mathcal{L}_{\text{rad}}&=D(r)\left(\frac{d^2}{dr^2}+\frac{k-1}{r}\frac{d}{dr}\right)+Q(r)\frac{d}{dr}+R(r),\\ \mathcal{L}_{\text{rad}}^l&=D(r)\left(\frac{d^2}{dr^2}+\frac{k-1}{r}\frac{d}{dr}-\frac{l^2}{r^2}\right)+R(r).\end{aligned} \] They show that the Fredholm index of either \(\mathcal{L}_{\text{rad}}\) or \(\mathcal{L}_{\text{rad}}^l\) is related to the Morse index of the operator at infinity under some suitable assumptions and, as a consequence, linear operators are Fredholm in exponentially weighted spaces for almost all weights. The result provides a basic tool for the analysis of perturbation and bifurcation problems in the presence of essential spectrum. A simple example is given to illustrate the application of the main results.