TBA equations for the Schrödinger equation with a regular singularity

Abstract

We derive the Thermodynamic Bethe Ansatz (TBA) equations for the Schrödinger equation with an arbitrary polynomial potential and a regular singular (simple and double pole) term. The TBA equations provide a non-trivial generalization of the ODE/IM correspondence and also give a solution for the Riemann-Hilbert problem in the exact WKB method. We study the TBA equations in detail for the linear and the harmonic oscillator potentials together with inverse and centrifugal terms. As an application, we also compute numerically the Voros spectrum for these potentials using the Bohr-Sommerfeld quantization condition.

39 pages, typos corrected

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• WKB periods for higher order ODE and TBA equations
• Wall-crossing of TBA equations and WKB periods for the third order ODE
• ODE/IM correspondence for affine Lie algebras: A numerical approach
• TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics
• TBA-like equations for non-planar scattering amplitude/Wilson lines duality at strong coupling
• Exact WKB Analysis and TBA Equations for the Stark Effect
• TBA equations and quantum periods for D-type Argyres-Douglas theories
• Thermodynamic Bethe ansatz and wall crossing for deformed supersymmetric quantum mechanics
• Nonperturbative Formulation of Resonances in Quantum Mechanics Based on Exact WKB Method
• Exact WKB in all sectors I: Potentials with degenerate saddles
• ODE/IQFT correspondence for the generalized affine $\mathfrak{sl}(2)$ Gaudin model
• Exact-WKB Analysis of Two-level Floquet Systems
• A short review on TBA equation and scattering amplitude/Wilson loop duality
• Exact WKB in all sectors II: Potentials with non-degenerate saddles