20 research outputs found
SuperintegrabilitÃ
Le simmetrie sono un ingrediente fondamentale per arrivare alla formulazione di leggi fisiche ed è possibile metterle in corrispondenza con quantità che si conservano e per tale ragione emergono nei sistemi più stabili e lontani dal caos. Tutti i sistemi che possono essere risolti per via analitica si dicono integrabili, tuttavia tra questi sistemi è possibile classificarne alcuni che hanno un numero di simmetrie massimale. Benché rari questi sistemi giocano un ruolo fondamentale dalla meccanica celeste alla fisica atomic
A maximally superintegrable deformation of the N-dimensional quantum Kepler–Coulomb system
XXIst International Conference on Integrable Systems and Quantum Symmetries (ISQS21,) 12–16 June 2013, Prague, Czech RepublicThe N-dimensional quantum Hamiltonian
Hˆ = −
~
2
|q|
2(η + |q|)
∇
2 −
k
η + |q|
is shown to be exactly solvable for any real positive value of the parameter η. Algebraically,
this Hamiltonian system can be regarded as a new maximally superintegrable η-deformation
of the N-dimensional Kepler–Coulomb Hamiltonian while, from a geometric viewpoint, this
superintegrable Hamiltonian can be interpreted as a system on an N-dimensional Riemannian
space with nonconstant curvature. The eigenvalues and eigenfunctions of the model are explicitly
obtained, and the spectrum presents a hydrogen-like shape for positive values of the deformation
parameter η and of the coupling constant k
New superintegrable models with position-dependent mass from Bertrand's Theorem on curved spaces
A generalized version of Bertrand's theorem on spherically symmetric curved
spaces is presented. This result is based on the classification of
(3+1)-dimensional (Lorentzian) Bertrand spacetimes, that gives rise to two
families of Hamiltonian systems defined on certain 3-dimensional (Riemannian)
spaces. These two systems are shown to be either the Kepler or the oscillator
potentials on the corresponding Bertrand spaces, and both of them are maximally
superintegrable. Afterwards, the relationship between such Bertrand
Hamiltonians and position-dependent mass systems is explicitly established.
These results are illustrated through the example of a superintegrable
(nonlinear) oscillator on a Bertrand-Darboux space, whose quantization and
physical features are also briefly addressed.Comment: 13 pages; based in the contribution to the 28th International
Colloquium on Group Theoretical Methods in Physics, Northumbria University
(U.K.), 26-30th July 201
Higher Order Quantum Superintegrability: a new "Painlev\'e conjecture"
We review recent results on superintegrable quantum systems in a
two-dimensional Euclidean space with the following properties. They are
integrable because they allow the separation of variables in Cartesian
coordinates and hence allow a specific integral of motion that is a second
order polynomial in the momenta. Moreover, they are superintegrable because
they allow an additional integral of order . Two types of such
superintegrable potentials exist. The first type consists of "standard
potentials" that satisfy linear differential equations. The second type
consists of "exotic potentials" that satisfy nonlinear equations. For , 4
and 5 these equations have the Painlev\'e property. We conjecture that this is
true for all . The two integrals X and Y commute with the Hamiltonian,
but not with each other. Together they generate a polynomial algebra (for any
) of integrals of motion. We show how this algebra can be used to calculate
the energy spectrum and the wave functions.Comment: 23 pages, submitted as a contribution to the monographic volume
"Integrability, Supersymmetry and Coherent States", a volume in honour of
Professor V\'eronique Hussin. arXiv admin note: text overlap with
arXiv:1703.0975
SuperintegrabilitÃ
Le simmetrie sono un ingrediente fondamentale per arrivare alla formulazione di leggi fisiche ed è possibile metterle in corrispondenza con quantità che si conservano e per tale ragione emergono nei sistemi più stabili e lontani dal caos. Tutti i sistemi che possono essere risolti per via analitica si dicono integrabili, tuttavia tra questi sistemi è possibile classificarne alcuni che hanno un numero di simmetrie massimale. Benché rari questi sistemi giocano un ruolo fondamentale dalla meccanica celeste alla fisica atomic
A Family of Exactly Solvable Radial Quantum Systems on Space of Non-Constant Curvature with Accidental Degeneracy in the Spectrum
A novel family of exactly solvable quantum systems on curved space is presented. The family is the quantum version of the classical Perlick family, which comprises all maximally superintegrable 3-dimensional Hamiltonian systems with spherical symmetry. The high number of symmetries (both geometrical and dynamical) exhibited by the classical systems has a counterpart in the accidental degeneracy in the spectrum of the quantum systems. This family of quantum problem is completely solved with the techniques of the SUSYQM (supersymmetric quantum mechanics). We also analyze in detail the ordering problem arising in the quantization of the kinetic term of the classical Hamiltonian, stressing the link existing between two physically meaningful quantizations: the geometrical quantization and the position dependent mass quantization