TBA equations for the Schrödinger equation with a regular singularity
Katsushi Ito, Hongfei Shu
Abstract
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.
Topics & Concepts
MathematicsWKB approximationMathematical analysisSingularityHarmonic oscillatorPolynomialBethe ansatzMathematical physicsGeneralizationSpectrum (functional analysis)Quantization (signal processing)AnsatzIndependent equationExact solutions in general relativityIntegral equationInverseInverse scattering transformHarmonicSingular solutionInverse scattering problemLinear equationQuantum Mechanics and Non-Hermitian PhysicsSpectral Theory in Mathematical PhysicsMathematical functions and polynomials