A dynamical systems approach to WKB-methods: The eigenvalue problem for a single well potential

In this paper, we revisit the eigenvalue problem of the one-dimensional Schr{\"o}dinger equation for smooth single well potentials. In particular, we provide a new interpretation of the Bohr-Sommerfeld quantization formula. A novel aspect of our results, which are based on recent work of the authors on the turning point problem based upon dynamical systems methods, is that we cover all eigenvalues $E\in [0,\mathcal O(1)]$ and show that the Bohr-Sommerfeld quantitization formula approximates all of these eigenvalues (in a sense that is made precise). At the same time, we provide rigorous smoothness statements of the eigenvalues as functions of $\epsilon$. We find that whereas the small eigenvalues $E=\mathcal O(\epsilon)$ are smooth functions of $\epsilon$, the large ones $E=\mathcal O(1)$ are smooth functions of $n\epsilon \in[c_1,c_2],\,0<c_1<c_2<\infty$, and $0\le \epsilon^{1/3}\ll 1$; here $n\in \mathbb N_0$ is the index of the eigenvalues.