Corrigendum: Fourth order superintegrable systems separating in polar coordinates. I. Exotic potentials (J. Phys. A: Math. Theor. 50 495206)

Adrian M Escobar-Ruiz; J C López Vieyra; P Winternitz

doi:10.1088/1751-8121/aaf06e

Journal of Physics A: Mathematical and Theoretical

Corrigendum • The following article is Free article

Corrigendum: Fourth order superintegrable systems separating in polar coordinates. I. Exotic potentials (J. Phys. A: Math. Theor. 50 495206)

Adrian M Escobar-Ruiz¹, J C López Vieyra² and P Winternitz¹

Published 30 November 2018 • © 2018 IOP Publishing Ltd
Journal of Physics A: Mathematical and Theoretical, Volume 52, Number 1 Citation Adrian M Escobar-Ruiz et al 2019 J. Phys. A: Math. Theor. 52 019501 DOI 10.1088/1751-8121/aaf06e

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This is a correction for 2017 J. Phys. A: Math. Theor. 50 495206

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escobarr@crm.umontreal.ca

vieyra@nucleares.unam.mx

wintern@crm.umontreal.ca

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¹ Centre de Recherches Mathématiques, and Département de Mathématiques et de Statistique, Université de Montréal, C.P. 6128, succ. Centre-ville, Montréal (QC) H3C 3J7, Canada

² Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 México, D.F., Mexico

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Adrian M Escobar-Ruiz https://orcid.org/0000-0002-4606-5972

J C López Vieyra https://orcid.org/0000-0002-7212-0672

P Winternitz https://orcid.org/0000-0003-1671-1213

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Received 31 October 2018
Accepted 13 November 2018
Published 30 November 2018

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Method: Single-anonymous
Screened for originality? Yes

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The left-hand side of the original equation (27), page 9, must be multiplied by $6\,r^2$ . The right-hand side is unchanged. The correct formula is the following:

$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle 6\,r^2\,\Theta(\theta) =&~ 288\, \Big[ B_{{1}} \sin 2\,\theta - B_{{2}} \cos 2\,\theta - 8 D_{{1}} \,\sin 4\,\theta + 8 D_{{2}} \,\cos 4\,\theta \Big] \, S \nonumber \\& +120\, \Big[ 20 D_{{1}} \,\cos 4\,\theta + 20 D_{{2}} \,\sin 4\,\theta - B_{{1}} \cos 2\,\theta - B_{{2}} \sin 2\,\theta \Big] \, S' \nonumber \\& +60\, \Big[ B_{{1}} \sin 2\,\theta - B_{{2}} \cos 2\,\theta + 14 D_{{1}} \,\sin 4\,\theta - 14 D_{{2}} \,\cos 4\,\theta \Big] \, S'' \nonumber \\& -30\, \Big[ B_{{1}} \cos 2\,\theta + B_{{2}} \sin 2\,\theta + 4 D_{{1}}\,\cos 4\,\theta + 4 D_{{2}}\,\sin 4\,\theta \Big] \, S^{(3)} \nonumber \\& -3\, \Big[ B_{{1}} \sin 2\,\theta - B_{{2}} \cos 2\,\theta +2 D_{{1}} \,\sin 4\,\theta -2 D_{{2}} \,\cos 4\,\theta \Big]\, S^{(4)} \nonumber \\ & - r \Big( 96\, \Big[ A_4 \,\sin \theta + A_3 \,\cos \theta - 9\, C_1\, \sin 3\,\theta - 9\, C_2\, \cos 3\,\theta \Big] \, S \nonumber \\& -120\, \Big[ A_{{4}} \,\cos \theta - A_{{3}} \,\sin \theta + 9\,C_2\,\sin 3\,\theta - 9\,C_1\,\cos 3\,\theta \Big] \, S' \nonumber \\& +480\, \Big[ \,C_1 \,\sin 3\,\theta + C_2 \,\cos 3\,\theta \Big] \,S'' \nonumber \\ & +30\, \Big[ A_{{3}}\,\sin \theta -A_{{4}}\,\cos \theta + 3\,C_2 \,\sin 3\,\theta - 3\,C_1 \,\cos 3\,\theta \Big] \, S^{(3)} \nonumber \\ & - 6\,\Big[ A_{{4}} \,\sin \theta + A_{{3}} \,\cos \theta + C_1 \,\sin 3\,\theta + C_2 \,\cos 3\,\theta \Big] \, S^{(4)} \Big) \nonumber \\& +576\,{r}^{2} \Big[ D_{{2}} \cos 4\,\theta - D_{{1}} \sin 4\,\theta \Big] R \nonumber \\& +6\,{r}^{3} \Big[ 6\, r^3 \, \big( A_{{2}} \, \sin \theta + A_{{1}}\, \cos \theta \big) -4\,{r}^{2} \big( B_{{4}} \cos 2\,\theta - B_{{3}} \sin 2\,\theta \big) \nonumber \\& -r\, \big( A_{{4}} \,\sin \theta + A_{{3}} \,\cos \theta + 9\, C_1 \,\sin 3\,\theta + 9\, C_2 \,\cos 3\,\theta \big) \nonumber \\& -3 \,B_{{1}}\,\sin 2\,\theta +3 \,B_{{2}}\,\cos 2\,\theta +66 \, D_{{1}}\,\sin 4\,\theta -66 \, D_{{2}}\,\cos 4\,\theta \Big] \, R' \nonumber \\& +6\,{r}^{4} \Big[ 6\,{r}^{3}\big( A_{{2}} \,\sin \theta + A_{{1}} \,\cos \theta \big) - 4\,{r}^{2} \big( B_{{3}} \sin 2\,\theta - B_{{4}} \cos 2\,\theta \big) \nonumber \\& + r\,\big( A_{{4}}\,\sin \theta + A_{{3}}\,\cos \theta + 9 \, C_1 \,\sin 3\,\theta + 9 \, C_2 \,\cos 3\,\theta \big) \nonumber \\& +3\,( B_{{1}} \,\sin 2\,\theta - B_{{2}} \,\cos 2\,\theta -6\, D_{{1}} \,\sin 4\,\theta +6\, D_{{2}} \,\cos 4\,\theta) \Big] \, R'' \nonumber \\& +3\,{r}^{5} \Big[ 2\,{r}^{3}\big( A_{{2}} \,\sin \theta + A_{{1}} \,\cos \theta \big) + 4\,{r}^{2} \big( \, B_{{4}}\cos 2\,\theta - \, B_{{3}}\sin 2\,\theta \big) \nonumber \\& + 2\,r\, \big( A_{{4}} \sin \theta + A_{{3}}\,\cos \theta - 3\, C_1 \,\sin 3\,\theta - 3\, C_2 \,\cos 3\,\theta \big) \nonumber \\& -2( B_{{1}}\,\sin 2\,\theta - B_{{2}}\,\cos 2\,\theta +2 D_{{2}}\,\cos 4\,\theta -2 D_{{1}}\,\sin 4\,\theta) \ \Big] R^{(3)} \ . \nonumber \label{CCsep} \nonumber \end{align} \tag{ 1 }$

All the results and conclusions of the paper remain unchanged. Dividing both sides of the above formula by $6\,r^2$ and then differentiating it with respect to $r$ we obtain the equation (22), page 7, presented in J. Phys. A: Math. Theor. 51 455202.

Corrigendum: Fourth order superintegrable systems separating in polar coordinates. I. Exotic potentials (J. Phys. A: Math. Theor. 50 495206)

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