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This paper has been submitted for publication in EMS Surveys in Mathematical Sciences. To download a preprint of this paper just click here.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-2204896 (Miller). Any opinions, findings and conclusions or recomendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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Rigorous methods for Bohr-Sommerfeld quantization rules

Joanne Dong, Peter D. Miller, and Giorgio Young

Department of Mathematics, University of Michigan, Ann Arbor

Abstract:

In this work, we 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.