Interrelation among Solvable Potentials and Extensions of SWKB Quantization Condition

• URL: http://arxiv.org/abs/2507.22381v2
• Date: Fri, 01 Aug 2025 07:07:55 GMT
• Title: Interrelation among Solvable Potentials and Extensions of SWKB Quantization Condition
• Authors: Yuta Nasuda,
• Abstract summary: We derive extended forms of the SWKB quantization condition for certain classes of Natanzon potentials.<n>We conjecture about the implication of the exactness of the SWKB in relation to the classical formula.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The exactly solvable Schr\"{o}dinger 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.

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