TBA equations and resurgent Quantum Mechanics

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.

42 pages, Typos corrected, references are added, published version

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Cited by in corpus (53)

• Physical Resurgent Extrapolation
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• Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$
• Exact WKB methods in $SU(2)$ $N_f=1$
• Geometric aspects of the ODE/IM correspondence
• Integrable Kondo problems
• TBA Equations and Quantization Conditions
• Exact-WKB analysis for SUSY and quantum deformed potentials: Quantum mechanics with Grassmann fields and Wess-Zumino terms
• On the Correspondence between Surface Operators in Argyres-Douglas Theories and Modules of Chiral Algebra
• Bootstrapping Calabi-Yau Quantum Mechanics
• TBA equations for the Schrödinger equation with a regular singularity
• The resurgent structure of quantum knot invariants
• Quasinormal Modes of C-metric from SCFTs
• Exact WKB and the quantum Seiberg-Witten curve for 4d $N=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization
• Opers for higher states of quantum KdV models
• WKB periods for higher order ODE and TBA equations
• An analytic evaluation of gravitational particle production of fermions via Stokes phenomenon
• A geometric recipe for twisted superpotentials
• Quantum Seiberg-Witten curve and Universality in Argyres-Douglas theories
• Functional Equations and Separation of Variables for Exact g-Function
• Exact WKB analysis and TBA equations for the Mathieu equation
• Wall-crossing of TBA equations and WKB periods for the third order ODE
• Exact WKB analysis for $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture
• ODE/IM correspondence for affine Lie algebras: A numerical approach
• Resurgence, Painleve Equations and Conformal Blocks
• $QQ$-system and non-linear integral equations for scattering amplitudes at strong coupling
• Quantum Seiberg-Witten periods for $\mathcal{N}=2$ $SU(N_c)$ SQCD around the superconformal point
• Non-Perturbative String Theory from AdS/CFT
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics
• Quantum periods and TBA equations for $\mathcal{N}=2SU(2)N_f=2$ SQCD with flavor symmetry
• On Chebyshev Wells: Periods, Deformations, and Resurgence
• Resonances and PT symmetry in quantum curves
• TBA-like equations for non-planar scattering amplitude/Wilson lines duality at strong coupling
• Exact WKB Analysis and TBA Equations for the Stark Effect
• Resurgence and semiclassical expansion in two-dimensional large-$N$ sigma models
• Exact instanton transseries for quantum mechanics
• A Schrödinger Equation for Evolutionary Dynamics
• WKB analysis of the linear problem for modified affine Toda field equations
• Quantum Periods and TBA-like Equations for a Class of Calabi-Yau Geometries
• Thermodynamic Bethe ansatz and wall crossing for deformed supersymmetric quantum mechanics
• Resurgence in complex Chern-Simons theory at generic levels
• TBA equations and quantum periods for D-type Argyres-Douglas theories
• Exact WKB in all sectors I: Potentials with degenerate saddles
• Quantum Rajeev-Ranken model as an anharmonic oscillator
• Nonperturbative Formulation of Resonances in Quantum Mechanics Based on Exact WKB Method
• Exact quantization and analytic continuation
• Exact-WKB Analysis of Two-level Floquet Systems
• Integrals of motion in $W{E}_6$ CFT and the ODE/IM correspondence
• Exact WKB in all sectors II: Potentials with non-degenerate saddles
• A short review on TBA equation and scattering amplitude/Wilson loop duality