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arXiv · 1811.04812

TBA equations and resurgent Quantum Mechanics

Katsushi Ito·Marcos Mariño·Hongfei Shu
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Abstract

We derive a system of TBA equations governing the exact WKB periods in one-dimensional Quantum Mechanics with arbitrary polynomial potentials. These equations provide a generalization of the ODE/IM correspondence, and they can be regarded as the solution of a Riemann-Hilbert problem in resurgent Quantum Mechanics formulated by Voros. Our derivation builds upon the solution of similar Riemann-Hilbert problems in the study of BPS spectra in $\mathcal{N}=2$ gauge theories and of minimal surfaces in AdS. We also show that our TBA equations, combined with exact quantization conditions, provide a powerful method to solve spectral problems in Quantum Mechanics. We illustrate our general analysis with a detailed study of PT-symmetric cubic oscillators and quartic oscillators.

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Katsushi Ito, Marcos Mariño, Hongfei Shu. 2019-01-31. TBA equations and resurgent Quantum Mechanics. https://doi.org/10.1007/jhep01(2019)228

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