An algebraic method of constructing potentials for which the Schrödinger equation with position dependent mass can be solved exactly is presented. A general form of the generators of su(1,1) algebra has been employed with a unified approach to the problem. Our systematic approach reproduces a number of earlier results and also leads to some novelties. We show that the solutions of the Schrödinger equation with position dependent mass are free from the choice of parameters for position dependent mass. Two classes of potentials are constructed that include almost all exactly solvable potentials.PACS numbers:

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We note that all regular and semiregular polytopes in arbitrary dimensions can be obtained from the Coxeter-Dynkin diagrams. The vertices of a regular or semiregular polytope are the weights obtained as the orbit of the Coxeter-Weyl group acting on the highest weight representing a selected irreducible representation of the Lie group. This paper, in particular, deals with the determination of the vertices of the Platonic and Archimedean solids from the Coxeter diagrams A3, B3, and H3 in the context of the quaternionic representations of the root systems and the Coxeter-Weyl groups. We use Lie algebraic techniques in the derivation of vertices of the polyhedra and show that the polyhedra possessing the tetrahedral, octahedral, and icosahedral symmetries are related to the Coxeter-Weyl groups representing the symmetries of the diagrams of A3, B3, and H3, respectively. This technique leads to the determination of the vertices of all Platonic and Archimedean solids except two chiral polyhedra, snubcuboctahedron and snubicosidodecahedron.

600-cell {3, 3, 5} and 120-cell {5, 3, 3} four-dimensional dual polytopes relevant to quasicrystallography have been studied with the quaternionic representation of the Coxeter group W(H4). The maximal subgroups W(SU(5)):Z2 and W(H3) × Z2 of W(H4) play important roles in the analysis of cell structures of the dual polytopes. In particular, the Weyl–Coxeter group W(SU(4)) is used to determine the tetrahedral cells of the polytope {3, 3, 5}, and the Coxeter group W(H3) is used for the dodecahedral cells of {5, 3, 3}. Using the Lie algebraic techniques in terms of quaternions, we explicitly construct cell structures forming the vertices of the 4D polytopes.