A Quasi-Maximally Superintegrable System
147
Note that the eigenfunctions Φ n,l (r) are identically zero at r = ξ for each l ≥ 0.
At the origin the situation is different since they are zero for l = 0 and approach a
constant value for l = 0.
Remark 2 We point out that the spectrum cannot be written in terms of a single
combination of the quantum numbers n and l, so it is not fully degenerate. This is a
clear indication that the system is not maximally superintegrable. However, QuasiMaximal Superintegrability (QMS) is ensured due to the hyperspherical symmetry
inside the punctured (open) hyperball B N
ξ :={x j > 0|
N
j =1 x 2
j < ξ 2 }.
3 Concluding Remarks and Future Perspectives
In the paper we have presented an example of a dynamically confined system, i.e.
a system whose eigenfunctions are square integrable and whose energy spectrum
is discrete by virtue of the functional form of the metrics and of the potential, not
because of external boundary conditions. In this sense, it is much similar to the
harmonic oscillator or to the Sutherland model. An interesting feature is that the
system is exactly solvable, its eigenfunctions being expressed in terms of polynomials (up to an algebraic pre-factor), though not being maximally superintegrable,
but just quasi-maximally superintegrable. Actually, the radial system does not seem
to be amenable neither to an intrinsic Kepler nor to an intrinsic oscillator. Although
a deeper investigation on this delicate point is certainly needed, at the present stage
we do not expect extra dynamical symmetries of Laplace–Runge–Lenz or Demkov–
Fradkin type [3–6]. Of course, we do not claim that exact solvability and maximal
superintegrability are unrelated notions [7, 8]: we just claim that on this topic there
is still something that has to be better understood.
Notes and Comments I want to mention that most of the results described in this
paper have been obtained in collaboration with my former student Danilo Latini,
who got his Ph.D.a couple of years ago. Unfortunately at the moment he has not got
any position whatsoever.
References
1. V. Perlick, Bertrand spacetimes. Class. Quantum Grav. 9 1009–1021 (1992)
2. J. Bertrand, Theoreme relatif au mouvement d’un point attire vers un centre fixe. C. R. Acad.
Sci. Paris 77 (1873) 849–853
3. A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, Bertrand spacetimes as Kepler/oscillator
potentials. Class. Quantum Grav. 25, 165005 (2008)
4. A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, Hamiltonian systems admitting a RungeLenz vector and an optimal extension of Bertrand’s theorem to curved manifolds. Commun.
Math. Phys. 290, 1033–1049 (2009)
147
Note that the eigenfunctions Φ n,l (r) are identically zero at r = ξ for each l ≥ 0.
At the origin the situation is different since they are zero for l = 0 and approach a
constant value for l = 0.
Remark 2 We point out that the spectrum cannot be written in terms of a single
combination of the quantum numbers n and l, so it is not fully degenerate. This is a
clear indication that the system is not maximally superintegrable. However, QuasiMaximal Superintegrability (QMS) is ensured due to the hyperspherical symmetry
inside the punctured (open) hyperball B N
ξ :={x j > 0|
N
j =1 x 2
j < ξ 2 }.
3 Concluding Remarks and Future Perspectives
In the paper we have presented an example of a dynamically confined system, i.e.
a system whose eigenfunctions are square integrable and whose energy spectrum
is discrete by virtue of the functional form of the metrics and of the potential, not
because of external boundary conditions. In this sense, it is much similar to the
harmonic oscillator or to the Sutherland model. An interesting feature is that the
system is exactly solvable, its eigenfunctions being expressed in terms of polynomials (up to an algebraic pre-factor), though not being maximally superintegrable,
but just quasi-maximally superintegrable. Actually, the radial system does not seem
to be amenable neither to an intrinsic Kepler nor to an intrinsic oscillator. Although
a deeper investigation on this delicate point is certainly needed, at the present stage
we do not expect extra dynamical symmetries of Laplace–Runge–Lenz or Demkov–
Fradkin type [3–6]. Of course, we do not claim that exact solvability and maximal
superintegrability are unrelated notions [7, 8]: we just claim that on this topic there
is still something that has to be better understood.
Notes and Comments I want to mention that most of the results described in this
paper have been obtained in collaboration with my former student Danilo Latini,
who got his Ph.D.a couple of years ago. Unfortunately at the moment he has not got
any position whatsoever.
References
1. V. Perlick, Bertrand spacetimes. Class. Quantum Grav. 9 1009–1021 (1992)
2. J. Bertrand, Theoreme relatif au mouvement d’un point attire vers un centre fixe. C. R. Acad.
Sci. Paris 77 (1873) 849–853
3. A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, Bertrand spacetimes as Kepler/oscillator
potentials. Class. Quantum Grav. 25, 165005 (2008)
4. A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, Hamiltonian systems admitting a RungeLenz vector and an optimal extension of Bertrand’s theorem to curved manifolds. Commun.
Math. Phys. 290, 1033–1049 (2009)