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
$\Gamma>0$.