We analyze the recently proposed classical/quantum mechanical interpretation of Zernike system and establish its equivalence to the Higgs oscillator on sphere or pseudosphere (Lobachevsky plane). We show that the non-reality of the classical Zernike Hamiltonian is an insignificant artifact of imaginary gauge and can be eliminated with a canonical transformation. The quantum counterpart of this canonical transformation is a similarity transformation mapping the system to the quantum Higgs oscillator with integration measure depending on [Formula: see text] parameters. When [Formula: see text], it results in the Hermitian Hamiltonian describing a free particle on (pseudo)sphere, while deviation from this point leads to a pseudo-Hermitian system.
An alternative method for the description of quantum superintegrable systems in two dimensions through the use of quantum algebraic techniques is introduced. It is suggested that such systems can be described in terms of a generalized deformed oscillator, characterized by a structure function specific to the system. The energy eigenvalues corresponding to a state with finite-dimensional degeneracy can then be determined directly from the properties of the relevant structure function. The validity of the method is demonstrated in the case of the isotropic harmonic oscillator in a space with constant curvature. The method can be used for constructing the quantum versions of several classical superintegrable systems, the Holt potential being given as an example.
In this short review paper the detailed analysis of six two-dimensional quantum {\it superintegrable} systems in flat space is presented. It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero and Wolfes (equivalently,
G2 rational, or
I6) models, and the Tremblay-Turbiner-Winternitz (TTW) system with integer index
k. It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture (2001); they admit algebraic forms for the Hamiltonian and both integrals (all three can be written as differential operators with polynomial coefficients without a constant term), they have polynomial eigenfunctions with the invariants of the discrete symmetry group of invariance taken as variables, they have hidden (Lie) algebraic structure
g(k) with various
k, and they possess a (finite order) polynomial algebras of integrals. Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag. Each subspace coincides with the finite-dimensional representation space of the algebra
g(k) for a certain
k. In all presented cases the algebra of integrals is a 4-generated
(H,I1,I2,I12≡[I1,I2]) infinite-dimensional algebra of ordered monomials of degrees 2,3,4,5, which is a subalgebra of the universal enveloping algebra of the hidden algebra.
The Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere, H Ψ = E Ψ, with generic 3-parameter potential. Clearly, the polynomials are expansion coefficients for one eigenbasis of a symmetry operator L 2 of H in terms of an eigenbasis of another symmetry operator L 1 , but the exact relationship appears not to have been made explicit. We work out the details of the expansion to show, explicitly, how the polynomials arise and how the principal properties of these functions: the measure, 3-term recurrence relation, 2nd order difference equation, duality of these relations, permutation symmetry, intertwining operators and an alternate derivation of Wilson functions - follow from the symmetry of this quantum system. This paper is an exercise to show that quantum mechancal concepts and recurrence relations for Gausian hypergeometrc functions alone suffice to explain these properties; we make no assumptions about the structure of Wilson polynomial/functions, but derive them from quantum principles. There is active interest in the relation between multivariable Wilson polynomials and the quantum superintegrable system on the n-sphere with generic potential, and these results should aid in the generalization. Contracting function space realizations of irreducible representations of this quadratic algebra to the other superintegrable systems one can obtain the full Askey scheme of orthogonal hypergeometric polynomials. All of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so (3, C ) and e (2, C ). All of the polynomials produced are interpretable as quantum expansion coefficients. It is important to extend this process to higher dimensions.
We analyze symmetries of the integrable generalizations of Jaynes–Cummings and Dicke models associated with simple Lie algebras g and their reductive subalgebras gK [T. Skrypnyk, “Generalized n-level Jaynes-Cummings and Dicke models, classical rational r-matrices and nested Bethe ansatz,” J. Phys. A: Math. Theor. 41, 475202 (2008)]. We show that their symmetry algebras contain commutative subalgebras isomorphic to the Cartan subalgebras of g, which can be added to the commutative algebras of quantum integrals generated with the help of the quantum Lax operators. We diagonalize additional commuting integrals and constructed with their help the most general integrable quantum Hamiltonian of the generalized n-level many-mode Jaynes–Cummings and Dicke-type models using nested algebraic Bethe ansatz.
Dirac’s Generalized Hamiltonian Dynamics (GHD) is a purely classical formalism for systems having constraints: it incorporates the constraints into the Hamiltonian. Dirac designed the GHD specifically for applications to quantum field theory. In one of our previous papers, we redesigned Dirac’s GHD for its applications to atomic and molecular physics by choosing integrals of the motion as the constraints. In that paper, after a general description of our formalism, we considered hydrogenic atoms as an example. We showed that this formalism leads to the existence of classical non-radiating (stationary) states and that there is an infinite number of such states—just as in the corresponding quantum solution. In the present paper, we extend the applications of the GHD to a charged Spherical Harmonic Oscillator (SHO). We demonstrate that, by using the higher-than-geometrical symmetry (i.e., the algebraic symmetry) of the SHO and the corresponding additional conserved quantities, it is possible to obtain the classical non-radiating (stationary) states of the SHO and that, generally speaking, there is an infinite number of such states of the SHO. Both the existence of the classical stationary states of the SHO and the infinite number of such states are consistent with the corresponding quantum results. We obtain these new results from first principles. Physically, the existence of the classical stationary states is the manifestation of a non-Einsteinian time dilation. Time dilates more and more as the energy of the system becomes closer and closer to the energy of the classical non-radiating state. We emphasize that the SHO and hydrogenic atoms are not the only microscopic systems that can be successfully treated by the GHD. All classical systems of N degrees of freedom have the algebraic symmetries ON+1 and SUN, and this does not depend on the functional form of the Hamiltonian. In particular, all classical spherically symmetric potentials have algebraic symmetries, namely O4 and SU3; they possess an additional vector integral of the motion, while the quantal counterpart-operator does not exist. This offers possibilities that are absent in quantum mechanics.