Projectively equivalent 2-dimensional superintegrable systems with projective symmetries

Vollmer, Andreas

doi:10.1088/1751-8121/ab6fc5

“…The possibility of relating some complex potentials from the 58 two-dimensional superintegrable systems through projective equivalence has been discussed [34,35] and different types of normal form, Liouville, complex Liouville, and Dini associated with Jordan blocks have been examined. The methods developed in this paper and the related methods will have applications to those models with complex Liouville and Dini form.…”

Section: Discussionmentioning

confidence: 99%

Dynamical symmetry algebras of two superintegrable two-dimensional systems

Marquette¹,

Quesne²

2022

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

A complete classification of 2D superintegrable systems on two-dimensional conformally flat spaces has been performed over the years and 58 models, divided into 12 equivalence classes, have been obtained. We will re-examine two pseudo-Hermitian quantum systems $E_{8}$ and $E_{10}$ from such a classification by a new approach based on extra sets of ladder operators. Those extra ladder operators are exploited to obtain the generating spectrum algebra and the dynamical symmetry one. We will relate the generators of the dynamical symmetry algebra to the Hamiltonian, thus demonstrating that the latter can be written in an algebraic form. We will also link them to the integrals of motion providing the superintegrability property. This demonstrates how the dynamical symmetry algebra explains the symmetries. Furthermore, we will exploit those algebraic constructions to generate extended sets of states and give the action of the ladder operators on them. We will present polynomials of the Hamiltonian and the integrals of motion that vanish on some of those states, then demonstrating that the sets of states not only contain eigenstates, but also generalized states. Our approach provides a natural framework for such states.

show abstract

“…The possibility of relating some complex potentials from the 58 two-dimensional superintegrable systems through projective equivalence has been discussed [34,35] and different types of normal form, Liouville, complex Liouville, and Dini associated with Jordan blocks have been examined. The methods developed in this paper and the related methods will have applications to those models with complex Liouville and Dini form.…”

Section: Discussionmentioning

confidence: 99%

Dynamical symmetry algebra of two superintegrable two-dimensional systems

Marquette,

Quesne

2022

Preprint

0

View full text Add to dashboard Cite

A complete classification of 2D superintegrable systems on two-dimensional conformally flat spaces has been performed over the years and 58 models, divided into 12 equivalence classes, have been obtained. We will re-examine two pseudo-Hermitian quantum systems E 8 and E 10 from such a classification by a new approach based on extra sets of ladder operators. Those extra ladder operators are exploited to obtain the generating spectrum algebra and the dynamical symmetry one. We will relate the generators of the dynamical symmetry algebra to the Hamiltonian, thus demonstrating that the latter can be written in an algebraic form. We will also link them to the integrals of motion providing the superintegrability property. This demonstrates how the dynamical symmetry algebra explains the symmetries. Furthermore, we will exploit those algebraic constructions to generate extended sets of states and give the action of the ladder operators on them. We will present polynomials of the Hamiltonian and the integrals of motion that vanish on some of those states, then demonstrating that the sets of states not only contain eigenstates, but also generalized states. Our approach provides a natural framework for such states.

show abstract

“…The problem of (locally) characterising the projective structures that are metrisable was first studied in the work of R. Liouville [17] in 1889, but was solved only relatively recently by Bryant, Dunajski and Eastwood for the case of two dimensions [2]. Since then, there has been renewed interest in the problem, see [5,6,8,10,11,13,14,25,27] for related recent work.…”

Section: Introductionmentioning

confidence: 99%

Metrisability of projective surfaces and pseudo-holomorphic curves

Mettler

¹

2020

View full text Add to dashboard Cite

We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure $${\mathfrak {p}}$$ p and a volume form $$\sigma $$ σ on an oriented surface M equip the total space of a certain disk bundle $$Z\rightarrow M$$ Z → M with a pair $$(J_{{\mathfrak {p}}},{\mathfrak {J}}_{{\mathfrak {p}},\sigma })$$ ( J p , J p , σ ) of almost complex structures. A conformal structure on M corresponds to a section of $$Z\rightarrow M$$ Z → M and $${\mathfrak {p}}$$ p is metrisable by the metric g if and only if $$[g] : M \rightarrow Z$$ [ g ] : M → Z is a pseudo-holomorphic curve with respect to $$J_{{\mathfrak {p}}}$$ J p and $${\mathfrak {J}}_{{\mathfrak {p}},dA_g}$$ J p , d A g .

show abstract

Projectively equivalent 2-dimensional superintegrable systems with projective symmetries

Cited by 6 publications

References 53 publications

Dynamical symmetry algebras of two superintegrable two-dimensional systems

Dynamical symmetry algebras of two superintegrable two-dimensional systems

Dynamical symmetry algebra of two superintegrable two-dimensional systems

Metrisability of projective surfaces and pseudo-holomorphic curves

Contact Info

Product

Resources

About