Quantum Systems at The Brink. Existence and Decay Rates of Bound States at Thresholds; Helium

Dirk Hundertmark, Michal Jex, Markus Lange

Published 2019-08-13Version 1

Existence and decay rates of eigenfunctions for Schr\"odinger operators provide interesting and important questions in quantum mechanics. It is well known that for eigenvalues below the threshold of the essential spectrum eigenvectors exist and decay exponentially. However, the situation at the threshold is much more subtle. In the present paper we propose a new method how to address both problems. We show how to calculate upper decay rate bounds at the threshold explicitly. As an example of application we show that for helium atom the decay rate of eigenvalues at the threshold of essential spectrum behaves as $\exp\left(-C\sqrt{|x|_\infty}\right)$ where $|x|_\infty=\max\{|x_1|,|x_2|\}$.