 0416 George A. Hagedorn, Julio H. Toloza
 A TimeIndependent BornOppenheimer Approximation with Exponentially Accurate Error Estimates
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Jan 22, 04

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Abstract. We consider a simple moleculartype 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 BornOppenheimer
parameter $\epsilon$ by optimal truncation of the RayleighSchr\"odinger
series. That is, we construct $E_\epsilon$ and $\Psi_\epsilon$, such that
$\\Psi_\epsilon\\,=\,O(1)$ and
\[
\\,(H(\epsilon)\,\,E_\epsilon))\,\Psi_\epsilon\,\\
<\ \Lambda\,\exp\,\left(\,\,{\Gamma}/{\epsilon^2}\,\right),\qquad
\mbox{where}\quad \Gamma>0.
\]
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