Upper and Lower Bounds on Scattering Lengths for Static Potentials

New upper and lower bounds on scattering lengths for static potentials are presented. The derivation is based on complementary variational principles for Fredholm integral equations of the type $p\left(r\right)\mathrm{}\varphi \mathrm{}\left(r\right)=f\left(r\right)-\int K(r, r^{\prime })\varphi \left(r^{\prime }\right)d{r}^{\prime }$ with symmetric, positive-definite kernels.