 0563 George A. Hagedorn, Julio H. Toloza
 Exponentially Accurate Quasimodes for the TimeIndependent
BornOppenheimer Approximation on a OneDimensional Molecular System
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Feb 10, 05

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Abstract. We consider the eigenvalue problem for a onedimensional
moleculartype quantum Hamiltonian that has the form
\[
H(\epsilon)\ =\ \,\frac{\epsilon^4}2\,
\frac{\partial^2\phantom{i}}{\partial y^2}\ +\ h(y),
\]
where $h(y)$ is an analytic
family of selfadjoint operators that has an discrete,
nondegenerate electronic level ${\cal E}(y)$
for $y$ in some open subset of ${\mathbb R}$.
Near a local minimum of the electronic 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 an energy $E_\epsilon$ and
a wave function $\Xi_\epsilon$, such that
the $L^2$norm of $\Xi_\epsilon$ is ${\cal O}(1)$ and the $L^2$norm of
$(H(\epsilon)\,\,E_\epsilon)\,\Xi_\epsilon$ is bounded by\ \,
$\Lambda\,\exp\,\left(\,\,{\Gamma}/{\epsilon^2}\,\right)\ $ with
$\Gamma>0$.
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