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Using the associated Jacobi differential equation, we obtain exactly bound states of the generalization of Woods–Saxon potential with the negative energy levels based on the analytic approach. According to the supersymmetry approaches in quantum mechanics, we show that these bound states by four pairs of the first-order differential operators, represent four types of the laddering equations. Two types of these supersymmetry structures, suggest the derivation of algebraic solutions by two different approaches for the bound states.

The Glauber minimum-uncertainty coherent states with two variables for Landau levels, based on the representation of Weyl-Heisenberg algebra by two different modes, have been studied about four decades ago. Here, we introduce new two-variable coherent states with minimum uncertainty relationship for Landau levels in three different methods: the infinite unitary representation of su(1, 1) is realized in two different methods, first, by consecutive levels with the same energy gaps and also with the same value for z-angular momentum quantum number, then, by shifting z-angular momentum mode number by two units while the energy level remaining the same. Besides, for su(2), whether by lowest Landau levels or Landau levels with lowest z-angular momentum, just one finite unitary representation is introduced. Having constructed the generalized Klauder-Perelomov coherent states, for any of the three representations, we obtain their Glauber coherency by displacement operator of Weyl-Heisenberg algebra.

The purpose of this paper is to generalize fermionic coherent states for two-level systems described by pseudo-Hermitian Hamiltonian [1], to n-level systems. Central to this task is the expression of the coherent states in terms of generalized Grassmann variables.These kind of Grassmann coherent states satisfy bi-overcompleteness condition instead of over-completeness one, as it is reasonably expected because of the biorthonormality of the system. Choosing an appropriate Grassmann weight function resolution of identity is examined. Moreover Grassmannian coherent and squeezed states of deformed group SU q (2) for three level pseudo-Hermitian system are presented.

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.

Firstly, the solvability of some quantum models like Eckart and Rosen–Morse II are explained on the basis of the shape invariance theory. Then, two generalized types of the Klauder–Perelomov and Gazeau–Klauder coherent states are calculated for the models. By means of calculating the Mandel parameter, it is shown that the weight distribution function of the first type coherent states obeys the Poissonian and super-Poissonian statistics, however, the weight distribution function of the second type coherent states obeys the Poissonian and sub-Poissonian statistics.

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