  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.
