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 (Formula presented.) 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 (Formula presented.). We find that whereas the small eigenvalues (Formula presented.) are smooth functions of (Formula presented.), the large ones (Formula presented.) are smooth functions of (Formula presented.), and (Formula presented.); here (Formula presented.) is the index of the eigenvalues.