Quantum Systems at The Brink

Abstract: We present a method to calculate the asymptotic behavior of eigenfunctions of Schr\"odinger operators that also works at the threshold of the essential spectrum. It can be viewed as a higher order correction to the well-known WKB method which does need a safety distance to the essential spectrum. We illustrate its usefulness on examples of quantum particles in a potential well with a long-range repulsive term outside the well.