Quantum systems at the brink: helium-type systems

In the present paper we study two challenging problems for helium–type systems. Existence of eigenvalues at thresholds and the asymptotic behavior of the corresponding eigenfunctions. Since the usual methods for addressing these problems need a safety distance to the essential spectrum, they cannot be applied in critical cases, when an eigenvalue enters the continuum.
We develop a method to address both problems and derive sharp upper and lower bounds for the asymptotic behavior of the ground state of critical helium–type systems at the threshold of the essential spectrum. This is the first proof of the precise asymptotic behavior of the ground state for this benchmark problem in quantum chemistry. Moreover, our bounds describe precisely how the asymptotic decay of the ground state changes, when the system becomes critical. In addition, we show the existence of a ground state of this quantum critical system with a finite nuclear mass. Previously this had been known only in the Born–Oppenheimer approximation of infinite nuclear mass.