A New Approach to Analysis of 2D
Higher Order Quantum Superintegrable
Systems
Bjorn K. Berntson, Ian Marquette, and Willard Miller, Jr.
Abstract We revise a method by Kalnins et al. (J Phys A Math Theor 43:265205,
2010) for constructing a canonical form for symmetry operators of arbitrary order
for the Schrödinger eigenvalue equation H Ψ ≡ (Δ 2 + V )Ψ = EΨ on any 2D
Riemannian manifold, real or complex, that admits a separation of variables in
some orthogonal coordinate system. Most of this paper is devoted to describing
the method. Details will be provided elsewhere. As examples we revisit the
Tremblay and Winternitz derivation of the Painlevé VI potential for a third order
superintegrable flat space system that separates in polar coordinates and, as new
results, we show that the Painlevé VI potential also appears for a third order
superintegrable system on the 2-sphere that separates in spherical coordinates, as
well as a third order superintegrable system on the 2-hyperboloid that separates in
spherical coordinates and one that separates in horocyclic coordinates. The purpose
of this project is to develop tools for analysis and classification of higher order
superintegrable systems on any 2D Riemannian space, not just Euclidean space.
Keywords Quantum superintegrable systems · Painlevé VI equation ·
Weierstrass equation
B. K. Berntson
Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden
e-mail: bbernts@kth.se
I. Marquette
School of Mathematics and Physics, The University of Queensland, Brisbane, QLD, Australia
e-mail: i.marquette@uq.edu.au; https://smp.uq.edu.au/profile/211/ian-marquette
W. Miller, Jr. ()
School of Mathematics, University of Minnesota, Minneapolis, MN, USA
e-mail: mille003@umn.edu
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_10
111
Higher Order Quantum Superintegrable
Systems
Bjorn K. Berntson, Ian Marquette, and Willard Miller, Jr.
Abstract We revise a method by Kalnins et al. (J Phys A Math Theor 43:265205,
2010) for constructing a canonical form for symmetry operators of arbitrary order
for the Schrödinger eigenvalue equation H Ψ ≡ (Δ 2 + V )Ψ = EΨ on any 2D
Riemannian manifold, real or complex, that admits a separation of variables in
some orthogonal coordinate system. Most of this paper is devoted to describing
the method. Details will be provided elsewhere. As examples we revisit the
Tremblay and Winternitz derivation of the Painlevé VI potential for a third order
superintegrable flat space system that separates in polar coordinates and, as new
results, we show that the Painlevé VI potential also appears for a third order
superintegrable system on the 2-sphere that separates in spherical coordinates, as
well as a third order superintegrable system on the 2-hyperboloid that separates in
spherical coordinates and one that separates in horocyclic coordinates. The purpose
of this project is to develop tools for analysis and classification of higher order
superintegrable systems on any 2D Riemannian space, not just Euclidean space.
Keywords Quantum superintegrable systems · Painlevé VI equation ·
Weierstrass equation
B. K. Berntson
Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden
e-mail: bbernts@kth.se
I. Marquette
School of Mathematics and Physics, The University of Queensland, Brisbane, QLD, Australia
e-mail: i.marquette@uq.edu.au; https://smp.uq.edu.au/profile/211/ian-marquette
W. Miller, Jr. ()
School of Mathematics, University of Minnesota, Minneapolis, MN, USA
e-mail: mille003@umn.edu
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_10
111