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 $$ \mathcal{N} $$ = 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.