George A. Hagedorn, Julio H. Toloza
A Time--Independent Born--Oppenheimer Approximation with Exponentially Accurate Error Estimates
(77K, latex)

ABSTRACT.  We consider a simple molecular--type quantum system in which the 
nuclei have one degree of freedom and the electrons have two levels. 
The Hamiltonian has the form 
\[ 
H(\epsilon)\ =\ -\,\frac{\epsilon^4}2\, 
\frac{\partial^2\phantom{i}}{\partial y^2}\ +\ h(y), 
\] 
where $h(y)$ is a $2\times 2$ real symmetric matrix. Near a local 
minimum of an electron level ${\cal E}(y)$ that is not at a level crossing, 
we construct quasimodes 
that are exponentially accurate in the square of the Born--Oppenheimer 
parameter $\epsilon$ by optimal truncation of the Rayleigh--Schr\"odinger 
series. That is, we construct $E_\epsilon$ and $\Psi_\epsilon$, such that 
$\|\Psi_\epsilon\|\,=\,O(1)$ and 
\[ 
\|\,(H(\epsilon)\,-\,E_\epsilon))\,\Psi_\epsilon\,\|\ 
<\ \Lambda\,\exp\,\left(\,-\,{\Gamma}/{\epsilon^2}\,\right),\qquad 
\mbox{where}\quad \Gamma>0. 
\]