Exact WKB Analysis and TBA Equations for the Stark Effect

TL;DR

This paper applies exact WKB analysis and TBA equations to the Stark effect in hydrogen, deriving precise quantization conditions and exploring their modifications, with numerical validation of the results.

Contribution

It introduces a novel application of exact WKB and TBA equations to the Stark effect, including modifications of quantization conditions and wall-crossing analysis.

Findings

Exact Bohr-Sommerfeld quantization conditions proven for specific parameter regions.

Modified quantization conditions with suppressed contributions beyond certain parameters.

Numerical validation of quantum periods and resonant frequencies.

Abstract

We apply the exact WKB analysis to a couple of one-dimensional Schroedinger-type equations reduced from the Stark effect of hydrogen in a uniform electric field. By introducing Langer's modification and incorporating the Stokes graphs, we prove the exactness of the Bohr-Sommerfeld quantization conditions for the Borel-resummed quantum WKB periods in the specific parameter regions of the electric field intensity and magnetic quantum number. It is also found these quantization conditions get modified with an additional suppressed contribution when the parameters vary beyond the specific regions. We also present Thermodynamic Bethe Ansatz (TBA) equations governing the quantum periods in the absence of Langer's modification and discuss its wall-crossing and analytic continuation. Numerical calculations are conducted to compare the complex resonant frequencies from our quantization conditions against ones from the Riccati-Pade method, the TBA equations are also confirmed by comparing its expansions with all-order quantum periods.