Pseudo-orthogonal groups and integrable dynamical systems in two dimensions

The quantum dynamics of a particle in the Modified Pöschl-Teller potential is derived from the group SL(2, R) by applying a Group Approach to Quantization (GAQ). The explicit form of the Hamiltonian as well as the ladder operators is found in the enveloping algebra of this basic symmetry group. The present algorithm provides a physical realization of the non-unitary, finite-dimensional, irreducible representations of the SL(2, R) group. The non-unitarity manifests itself in that only half of the states are normalizable, in contrast with the representations of SU (2) where all the states are physical.

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“…The commutation relations in (12) and in (18) are formally analogous provided that we redefine in (18) the generatorX…”

Section: The Quantum Pöschl-teller Systemmentioning

confidence: 99%

Group approach to the quantization of the Pöschl–Teller dynamics

Aldaya¹,

Guerrero

2005

J. Phys. A: Math. Gen.

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“…where α 2 = E and α 1 are the separation constants (which are positive). Each one of these two equations is formaly similar to those of the corresponding one-dimensional problem [13]. The solutions of both HJ equations are easily computed and can be found as particular cases in [12].…”

Section: The Hamilton-jacobi Equation For the U(3)-systemmentioning

confidence: 99%

“…If we consider together all the IO's {Â ± ,Â,B ± ,B,Ĉ ± ,Ĉ} we find that they close a su(2, 1) Lie algebra, whose Lie commutators are displayed in (9), (13) and (14) together with the crossed commutators…”

Section: Algebraic Structure Of the Intertwining Operatorsmentioning

confidence: 99%

Intertwining Symmetry Algebras of Quantum Superintegrable Systems

Calzada

Negro

Olmo

2009

SIGMA

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Abstract. We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwinig operators, that span pairs of Lie algebras like (su(n), so(2n)) or (su(p, q), so(2p, 2q)). The eigenstates of the associated Hamiltonian hierarchies belong to unitary representations of these algebras. It is shown that these intertwining operators, related with separable coordinates for the system, are very useful to determine eigenvalues and eigenfunctions of the Hamiltonians in the hierarchy. An study of the corresponding superintegrable classical systems is also included for the sake of completness.

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“…When there is more than (N − 1) integrals of motion (not all of them in involution), the system is called superintegrable. Superintegrablity is also closely related to the fact that the Hamilton-Jacobi (H-J) equation is separable in more than one coordinate system [10]. Izmest'ev et al [12] showed that the separable coordinates of free systems on the two-sphere can be turned into the separable coordinates on the Euclidean plane by using contractions.…”

Section: Introductionmentioning

confidence: 98%