Semi-classical quantization of three-particles Toda lattice improved

The conventional methods for quantizing the three-particle Toda system typically involve complex numerical procedures that require multiple steps, often becoming computationally intensive to ensure accuracy and reliability. In this work, we aim to streamline the Einstein–Brillouin–Keller (EKB) quantization process by minimizing the reliance on numerical computations. To achieve this, we present a reformulation of the mathematical framework using a new set of more suitable variables. These variables not only simplify the structure of the problem but also reduce or eliminate certain removable singularities and divergences that commonly arise in the standard approach. As a result, the numerical integration needed for determining the semi-classical energy levels is significantly simplified, leading to a more efficient and less time-consuming procedure. This reformulation offers a more practical approach to quantizing the three-particle Toda system, while retaining the accuracy and precision required in semi-classical analyses.