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{ö}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 $n\epsilon \in [c_1,c_2],0<c_1<c_2<\mathrm{\infty }$, and $0\le {\epsilon }^{1\mathrm{/}3}\ll 1$; here $n\in {\mathbb{N}}_0$ is the index of the eigenvalues.