05-63 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) Feb 10, 05
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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 $\Gamma>0$.

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