SWKB Quantization Condition for Conditionally Exactly Solvable Systems and the Residual Corrections

The SWKB quantization condition is an exact quantization condition for the conventional shape-invariant potentials. On the other hand, this condition equation does not hold for other known solvable systems. The origin of the (non-)exactness is understood in the context of the quantum Hamilton--Jacobi formalism. First, we confirm the statement and show inexplicit properties numerically for the case of the conditionally exactly solvable systems by Junker and Roy. The SWKB condition breaks for this case, but the condition equation is restored within a certain degree of accuracy. We propose a novel approach to evaluate the residual by perturbation, intending to explore the correction terms for the SWKB condition equation.