dorsal/arxiv
View SchemaExactness of Conventional and Supersymmetric JWKB Formulae and Glo bal Symmetries of Stokes Graphs
| Authors | Piotr Milczarski |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9807039 |
| URL | https://arxiv.org/abs/quant-ph/9807039 |
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.
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"abstract": "It has been shown that the cases of the JWKB formulae in 1--dim QM quantizing\nthe energy levels exactly are results of essentially one global symmetry of\nboth potentials and their corresponding Stokes graphs. Namely, this is the\ninvariance of the latter on translations in the complex plain of the space\nvariable i.e. the potentials and the Stokes graphs have to be periodic. A\nproliferation of turning points in the basic period strips (parallelograms) is\nanother limitation for the exactness of the JWKB formulae. A systematic\nanalyses of a single-well class of potentials satisfying suitable conditions\nhas been performed. Only ten potentials (with one or two real parameters)\nquantized exactly by the JWKB formulae have been found all of them coinciding\n(or being equivalent to) with the well-known ones found previously. It was\nshown also that the exactness of the supersymmetric JWKB formulae is a\nconsequence of the corresponding exactness of the conventional ones and vice\nversa. Because of the latter two exactly JWKB quantized potentials have been\nadditionally established. These results show that the exact SUSY JWKB formulae\nchoose the Comtet at al form of them independently of whether the supersymmetry\nis broken or not. A close relation between the shape invariance property of\npotentials considered and their meromorphic structure on the x-plane is also\ndemonstrated.",
"arxiv_id": "quant-ph/9807039",
"authors": [
"Piotr Milczarski"
],
"categories": [
"quant-ph"
],
"title": "Exactness of Conventional and Supersymmetric JWKB Formulae and Glo bal Symmetries of Stokes Graphs",
"url": "https://arxiv.org/abs/quant-ph/9807039"
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