Exactness of Conventional and Supersymmetric JWKB Formulae and Glo bal Symmetries of Stokes Graphs

TL;DR

This paper investigates the conditions under which JWKB formulae exactly quantize energy levels in quantum mechanics, revealing that global symmetries and potential periodicity are key, and explores the relationship between supersymmetry and exact quantization.

Contribution

It systematically identifies all single-well potentials exactly quantized by JWKB, linking their exactness to global symmetries and supersymmetry, and clarifies the role of shape invariance and meromorphic structure.

Findings

Exactly JWKB quantized potentials are linked to potential periodicity and symmetry.

Supersymmetric JWKB exactness follows from conventional JWKB exactness.

Two additional potentials are identified that are exactly quantized by JWKB.

Abstract

It has been shown that the cases of the JWKB formulae in 1--dim QM quantizing the energy levels exactly are results of essentially one global symmetry of both potentials and their corresponding Stokes graphs. Namely, this is the invariance of the latter on translations in the complex plain of the space variable i.e. the potentials and the Stokes graphs have to be periodic. A proliferation of turning points in the basic period strips (parallelograms) is another limitation for the exactness of the JWKB formulae. A systematic analyses of a single-well class of potentials satisfying suitable conditions has been performed. Only ten potentials (with one or two real parameters) quantized exactly by the JWKB formulae have been found all of them coinciding (or being equivalent to) with the well-known ones found previously. It was shown also that the exactness of the supersymmetric JWKB formulae is a consequence of the corresponding exactness of the conventional ones and vice versa. Because of the latter two exactly JWKB quantized potentials have been additionally established. These results show that the exact SUSY JWKB formulae choose the Comtet at al form of them independently of whether the supersymmetry is broken or not. A close relation between the shape invariance property of potentials considered and their meromorphic structure on the x-plane is also demonstrated.