TBA equations and resurgent Quantum Mechanics

• Main Authors: Katsushi Ito, Marcos Mariño, Hongfei Shu
• Format: Article
• Language: English
• Published: SpringerOpen 2019-01-01
• Series: Journal of High Energy Physics
• Subjects: Integrable Field Theories; Nonperturbative Effects; Topological Field Theories
• Online Access: http://link.springer.com/article/10.1007/JHEP01(2019)228

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 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.