TBA Equations and Quantization Conditions

Abstract

It has been recently realized that, in the case of polynomial potentials, the exact WKB method can be reformulated in terms of a system of TBA equations. In this paper we study this method in various examples. We develop a graphical procedure due to Toledo, which provides a fast and simple way to study the wall-crossing behavior of the TBA equations. When complemented with exact quantization conditions, the TBA equations can be used to solve spectral problems exactly in Quantum Mechanics. We compute the quantum corrections to the all-order WKB periods in many examples, as well as the exact spectrum for many potentials. In particular, we show how this method can be used to determine resonances in unbounded potentials.

arXiv comment77 pages, 13 figures; typos corrected

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• Exact WKB analysis for $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture
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• Exact WKB Analysis and TBA Equations for the Stark Effect
• Nonperturbative Formulation of Resonances in Quantum Mechanics Based on Exact WKB Method
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• Exact WKB in all sectors I: Potentials with degenerate saddles
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• Exact WKB in all sectors II: Potentials with non-degenerate saddles