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.
Highlights
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
We show that our TBA equations, combined with exact quantization conditions, provide a powerful method to solve spectral problems in Quantum Mechanics
This correspondence was originally based on functional relations discovered in the context of resurgent Quantum Mechanics, which are similar to Baxter-type equations appearing in integrable systems
Summary
We give a lightning review of the exact WKB method from the point of view of the theory of resurgence. In the exact WKB method, the quantum periods, which start their life as formal power series in , are promoted to actual functions by the procedure of Borel resummation. It can be analytically continued to a function on the complex plane, displaying in general various types of singularities (typically poles and branch cuts.) The Borel resummation of the quantum period is defined by a Laplace transform, s (Πγ) ( ) = 1. The structure of discontinuities in Quantum Mechanics involves just the quantum periods on the WKB curve, and no other formal power series intervene. In calculating the quantum period associated to the cycle γ2 around the forbidden interval, we have to make a choice of branch cut of the momentum function p(q) The choice is such that iΠγ2 is real and positive, the discontinuity appearing in the r.h.s. of (2.20) is exponentially small.
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