04-16 George A. Hagedorn, Julio H. Toloza
A Time--Independent Born--Oppenheimer Approximation with Exponentially Accurate Error Estimates (77K, latex) Jan 22, 04
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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 \[ \|\,(H(\epsilon)\,-\,E_\epsilon))\,\Psi_\epsilon\,\|\ <\ \Lambda\,\exp\,\left(\,-\,{\Gamma}/{\epsilon^2}\,\right),\qquad \mbox{where}\quad \Gamma>0. \]

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