A Dynamical Systems Approach to WKB‐Methods: The Eigenvalue Problem for a Single Well Potential

ABSTRACT In this paper, we revisit the eigenvalue problem of the one‐dimensional Schrödinger 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 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 . We find that whereas the small eigenvalues are smooth functions of , the large ones are smooth functions of , and ; here is the index of the eigenvalues.