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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

classification hep-th
keywords equationsequationodingerpotentialsregularschransatzapplication
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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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