Interrelation among Solvable Potentials and Extensions of SWKB Quantization Condition

The exactly solvable Schrödinger equations with the conventional shape-invariant potentials are known to be related with each other through point cannonical transformations. In this paper, we extend the idea to integral formulae called the SWKB integrals. By virtue of this, we derive extended forms of the SWKB quantization condition for certain classes of Natanzon potentials. We further demonstrate that the same idea can also be applied to obtain an exact quantization rule for a subclass of quantum systems with position-dependent effective masses, provided their solutions involve the classical orthogonal polynomials. Based on the findings, we conjecture about the implication of the exactness of the SWKB formula in relation to the classical orthogonal polynomials.