Title: Quantum Systems at the Brink: Helium
Abstract: Perturbation theoryworks well for the the discrete spectrum below the essential spectrum. Whathappens if a parameter of a quantum system is tuned in such a way that abound state energy (e.g. the ground state energy) hits the bottom of theessential spectrum? Does the eigenvalue survive, i.e., the correspondingeigenfunction stays $L^2$, or does it dissolve into the continuumenergies?
It turns out the
a) The answer depends on thedimension and
b) a long range repulsive part ofthe which classically should push the system apart, will stabilize thequantum system (This might sound weird, but I'll explain it :-))
c) In particular, a long-rangerepulsive part of the potential helps to get WKB type decay estimates for boundstates whose eigenvalues are at the bottom of the essential spectrum, whenthe usual approach due to Agmon does not apply.
I will explain the main ideas andtricks on how one can get upper bounds on the decay of eigenfunction when theusual Agmon method does not work, in a one particle toy model and thecase of helium at critical coupling.
The methods apply to a variety ofcritical quantum systems, where an eigenvalue enters the continuum, but staysan eigenvalue.This is joint work with MarkusLange (UBC) and Michal Jex (Prague).