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TBA equations for the Schr\"odinger equation with a regular singularity

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arxiv 1910.09406 v2 pith:FV7UYX6M submitted 2019-10-21 hep-th

Katsushi Ito , Hongfei Shu This is my paper
classification hep-th
keywords equationsequationodingerpotentialsregularschransatzapplication
verification ladder T0 review T1 audit T2 compute T3 formal
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We derive the Thermodynamic Bethe Ansatz (TBA) equations for the Schr\"odinger 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.

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Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. TBA equations for $SU(r+1)$ quantum Seiberg-Witten curve: higher-order Mathieu equation

    hep-th 2026-04 unverdicted novelty 7.0 of 10

    TBA equations are derived for the higher-order Mathieu equation of the SU(r+1) quantum Seiberg-Witten curve, with an analytic effective central charge and subleading agreement with the WKB method.

  2. Exact WKB analysis of inverted triple-well: resonance, PT-symmetry breaking, and resurgence

    hep-th 2026-04 unverdicted novelty 7.0 of 10

    Exact WKB analysis of inverted triple-well potential reveals PT-symmetry breaking at an exceptional point given by a simple relation between bounce and bion actions, with median-summed spectra real or complex accordingly.

  3. Exact WKB analysis of inverted triple-well: resonance, PT-symmetry breaking, and resurgence

    hep-th 2026-04 unverdicted novelty 7.0 of 10

    Exact WKB analysis produces median-summed spectra and an algebraic equation for the exceptional point of PT-symmetry breaking in the inverted triple-well system.

  4. Exact WKB in all sectors II: Potentials with non-degenerate saddles

    hep-th 2025-11 conditional novelty 7.0 of 10

    For generic one-dimensional potentials, the exact spectrum decomposes into as many trans-series sectors as there are distinct local-minimum energy levels, with continuous transitions across barrier tops and discontinu...

  5. From classical Lax ODEs to quantum integrable theories: the moduli

    hep-th 2026-05 unverdicted novelty 6.0 of 10

    The paper derives moduli-modified functional relations for Wronskians of a classical Lax ODE that identify quantum states, produce Y-systems and TBA equations without scattering theory, and prove two Zamolodchikov con...

  6. Exact WKB and Quantum Periods for Extremal Black Hole Quasinormal Modes

    hep-th 2026-05 unverdicted novelty 6.0 of 10

    Exact WKB with high-order quantum period computations and Borel-Padé resummation reproduces quasinormal mode frequencies for extremal Reissner-Nordström and Kerr black holes.

  7. TBA equations for $SU(r+1)$ quantum Seiberg-Witten curve: higher-order Mathieu equation

    hep-th 2026-04 unverdicted novelty 6.0 of 10

    Derives TBA equations for the higher-order Mathieu equation of the SU(r+1) quantum Seiberg-Witten curve, obtains an analytic effective central charge from Y-function boundary conditions at theta to -infinity, and veri...

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