TBA equations and resurgent Quantum Mechanics

Katsushi Ito (Department of Physics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan) ; Marcos Mariño (Département de Physique Théorique & Section de Mathématiques, Université de Genève, Genève, CH-1211, Switzerland) ; Hongfei Shu (Department of Physics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan)

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 N $$ \mathcal{N} $$ = 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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      "surname": "Ito", 
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          "organization": "Universit\u00e9 de Gen\u00e8ve"
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      "surname": "Mari\u00f1o", 
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      "value": "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   <math> <mi>N</mi> </math>  $$ \\mathcal{N} $$  = 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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Published on:
30 January 2019
Publisher:
Springer
Published in:
Journal of High Energy Physics , Volume 2019 (2019)
Issue 1
Pages 1-45
DOI:
https://doi.org/10.1007/JHEP01(2019)228
Copyrights:
The Author(s)
Licence:
CC-BY-3.0

Fulltext files: