George A. Hagedorn, Julio H. Toloza
Exponentially Accurate Quasimodes for the Time--Independent 
Born--Oppenheimer Approximation on a One--Dimensional Molecular System
(67K, LaTeX 2e)

ABSTRACT.  We consider the eigenvalue problem for a one-dimensional 
molecular--type 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 self-adjoint 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 Born--Oppenheimer 
parameter $\epsilon$ by optimal truncation of the Rayleigh--Schr\"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 
$\Gamma>0$.