A Confined Quasi-Maximally
Superintegrable N -dimensional System,
Classical and Quantum, in a Space
with Variable Curvature
Orlando Ragnisco
Abstract In the present paper I will briefly summarize some recent results about the
solvability of the classical and quantum version of a (hyper-)spherically symmetric
N-dimensional system living on a curved manifold characterized by a conformally
flat metric. The system appears as a generalization of the so-called Taub–NUT
system. We call it Quasi-Maximally Superintegrable (QMS) since it is endowed with
2N − 2 constants of the motion (with 2N − 1 it would have been Maximally Superintegrable (MS)) functionally independent and Poisson commuting in the Classical
case, algebraically independent and commuting as operators in the Quantum case.
The eigenvalues and eigenfunctions of the quantum system are explicitly given,
while for the classical version we provide the analytic solution of the radial equation
of motion. A few comments about the connection between exact solvability and
superintegrability are made in the final part of the paper.
Keywords Hamiltonian systems · Superintegrability · Exact solvability
1 The Classical Model: General Features
We consider the two-parameter family of N-dimensional (N D) classical Hamiltonian systems (a degenerate case of the Perlick’s II system, introduced by V. Perlick
in the seminal paper [1], where a highly nontrivial extension of Bertrand theorem
[2] to curved spacetimes was proven), given by
H = T (q, p) + U (q) =
1 −
|q|
ξ
|q|
ξ
p 2
2
+
k
ξ − |q|
(1)
O. Ragnisco ()
Roma TRE University, Rome, Italy
e-mail: ragnisco@fis.uniroma3.it; http://webusers.fis.uniroma3.it/~ragnisco
© 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_13
141
Superintegrable N -dimensional System,
Classical and Quantum, in a Space
with Variable Curvature
Orlando Ragnisco
Abstract In the present paper I will briefly summarize some recent results about the
solvability of the classical and quantum version of a (hyper-)spherically symmetric
N-dimensional system living on a curved manifold characterized by a conformally
flat metric. The system appears as a generalization of the so-called Taub–NUT
system. We call it Quasi-Maximally Superintegrable (QMS) since it is endowed with
2N − 2 constants of the motion (with 2N − 1 it would have been Maximally Superintegrable (MS)) functionally independent and Poisson commuting in the Classical
case, algebraically independent and commuting as operators in the Quantum case.
The eigenvalues and eigenfunctions of the quantum system are explicitly given,
while for the classical version we provide the analytic solution of the radial equation
of motion. A few comments about the connection between exact solvability and
superintegrability are made in the final part of the paper.
Keywords Hamiltonian systems · Superintegrability · Exact solvability
1 The Classical Model: General Features
We consider the two-parameter family of N-dimensional (N D) classical Hamiltonian systems (a degenerate case of the Perlick’s II system, introduced by V. Perlick
in the seminal paper [1], where a highly nontrivial extension of Bertrand theorem
[2] to curved spacetimes was proven), given by
H = T (q, p) + U (q) =
1 −
|q|
ξ
|q|
ξ
p 2
2
+
k
ξ − |q|
(1)
O. Ragnisco ()
Roma TRE University, Rome, Italy
e-mail: ragnisco@fis.uniroma3.it; http://webusers.fis.uniroma3.it/~ragnisco
© 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_13
141