The phenomenon of localization, i.e. the appearance
of intervals of dense point spectrum associated with exponentially decaying
eigenfunctions, has become a popular topic in the spectral theory of Schrödinger
operators. The antithetical event, namely isolated eigenvalues with long range
eigenfunctions, is not so well-known. There are examples for the one-dimensional
case which provide insight into the limitations of some theorems relating the
behavior of eigensolutions at infinity to the spectrum. Such ideas have also been employed to
show that Dirac operators with potentials tending to infinity may have holes in the
essential spectrum. We will summarize these results and
indicate possible approaches for higher dimensions.