Binding energy for hydrogen-like atoms in the Nelson model without cutoffs

Hainzl, Christian; Hirokawa, Masao; Spohn, Herbert

Binding energy for hydrogen-like atoms in the Nelson model without cutoffs

Authors

Hainzl, Christian

Hirokawa, Masao

Spohn, Herbert

Description

In the Nelson model particles interact through a scalar massless field. For hydrogen-like atoms there is a nucleus of infinite mass and charge $Ze$, $Z > 0$, fixed at the origin and an electron of mass $m$ and charge $e$. This system forms a bound state with binding energy $E_{\rm bin} = me^4Z^2/2$ to leading order in $e$. We investigate the radiative corrections to the binding energy and prove upper and lower bounds which imply that $ E_{\rm bin} = me^4 Z^2/2 + c_0 e^6 + \Ow(e^7 \ln e)$ with explicit coefficient $c_0$ and independent of the ultraviolet cutoff. $c_0$ can be computed by perturbation theory, which however is only formal since for the Nelson Hamiltonian the smallest eigenvalue sits exactly at the bottom of the continuous spectrum.
30 pages, AMSLatex