In the previous chapter, we observed a remarkable and mysterious relation between the Schrödinger equation and the quantum integrable model. In this chapter, we will explore a more detailed structure of the relations based on the exact WKB analysis for the Schrödinger equation with a polynomial potential term. The WKB approximation is a semiclassical method of calculating the wave function that is assumed to be an exponential form, where its amplitude and phase change slowly against the coordinate. The wave function is expanded in the Planck constant \(\hbar \) , which provides an asymptotic series. The Borel resummation, which transforms the asymptotic series into an analytic function, is a useful method to handle the wave function and its analytic continuation exactly.

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Exact WKB Analysis and TBA Equations

  • Katsushi Ito,
  • Hongfei Shu

摘要

In the previous chapter, we observed a remarkable and mysterious relation between the Schrödinger equation and the quantum integrable model. In this chapter, we will explore a more detailed structure of the relations based on the exact WKB analysis for the Schrödinger equation with a polynomial potential term. The WKB approximation is a semiclassical method of calculating the wave function that is assumed to be an exponential form, where its amplitude and phase change slowly against the coordinate. The wave function is expanded in the Planck constant \(\hbar \) , which provides an asymptotic series. The Borel resummation, which transforms the asymptotic series into an analytic function, is a useful method to handle the wave function and its analytic continuation exactly.