Rigorous methods for Bohr–Sommerfeld quantization rules

Abstract

In this paper, we will prove Bohr–Sommerfeld quantization rules for the self-adjoint Zakharov–Shabat system and the Schrödinger equation in the presence of two simple turning points bounding a classically allowed region. In particular, we use the method of comparison equations for $2\times 2$ traceless first-order systems to provide a unified perspective that yields similar proofs in each setting. The use of a Weber model system gives results that are uniform in the eigenvalue parameter over the whole range from the bottom of the potential well up to finite values.