Master function approach to quantum solvable models on SL(2,c) and SL(2,c)/GL(1,c) manifolds

Abstract

de Rham cohomology of SO (n) and some related manifolds by supersymmetric quantum mechanics By introducing a new parameter as a second associated index for special functions, we construct the three-dimensional differential generators of gl(2,c) Lie algebra together with the corresponding contracted form h 4 . Non-Casimir quadratic as well as the Casimir of gl(2,c) ͑and h 4 ͒ generators can be considered as quantum solvable models on group manifold SL(2,c). Then, by appropriate parametrization of group manifold SL(2,c) and eliminating one of the coordinates, we obtain quantum solvable Hamiltonians on homogeneous manifold SL(2,c)/GL(1,c) with the metric described by master function. We show that two-dimensional Hamiltonian on SL(2,c)/GL(1,c) derived from the reduction of Casimir operator so(4,c) Lie algebra as a three-dimensional Hamiltonian on group manifold SL(2,c), possesses the degeneracy SL(2,c) group and, also, the shape invariance property, where both have para-supersymmetry representations of arbitrary order.

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“…relations (18) and (20) can lead to the satisfaction of (15b). Using the second equation of (16), the left-hand side of relation (15b) becomes…”

“…Considering relations (2) and (20) for dynamical symmetry groups U (2) and U(1, 1), it can be shown that in the cases p = 2 and 3 we have β 1 = 1 and β 1 = β 2 = 1 respectively, and consequently the extended PSUSY algebra is reduced to the RSK standard PSUSY. Thus in the cases u (2) and u(1, 1), the extension of PSUSY is automatically accomplished for p 4.…”

“…Furthermore, when a = 0, one have to accept constraint (20) again for the parameters β 1 , β 2 , . .…”

“…Indeed, it seems that only realization of relations (15a) and (15d) for a = 0 requires to accept constraint (20). On the other hand, one may check that the left-hand sides of relations (15a) and (15c) are Hermitian conjugate of each other as well as the left-hand sides of (15b) and (15d).…”

“…Afterwards, Khare generalized the PSUSY of order 2 to the arbitrary order p [15][16][17]. In the realization of PSUSY by the shape invariant quantum solvable models such as simple harmonic oscillator, three-dimensional oscillator, Morse, Scarf I, Scarf II, generalized Pöschl-Teller and Natanzon potentials [18] as well as the quantum solvable models with symmetry of real forms of gl (2, c) Lie algebra like the Landau levels problem on the flat surface with symmetry of Heisenberg Lie algebra h 4 , and the motion of a charged particle on the two-dimensional sphere in the presence or absence of a magnetic monopole [19,20], the Rubakov-Spiridonov-Khare (RSK) algebra has been much more successful than the Beckers and Debergh algebra. For this reason, the symmetry between bosons and parafermions of order p (with p = 1, 2, .…”