Abstract

DA ROCHA, R.  and  OLIVEIRA, E.Capelas de. The Casimir operator of SO(1,2) and the Pöschl-Teller potential: an AdS approach. Rev. mex. fis. [online]. 2005, vol.51, n.1, pp.01-04. ISSN 0035-001X.

We present and discuss some features of the anti-de Sitter spacetime, that is jointly with de Sitter and Minkowski is only, the unique maximal isotropic manifold. Among all possible lorentzian manifolds, we restrict our attention to the anti-de Sitter (AdS) spacetime, with metric diag(1,-1, -1). We start by presenting the conformal time metric on AdS and we then show how we can obtain the Schrödinger formalism [1]. The Lie algebra so(1,2) is introduced and used to construct spin and ladder operators. After presenting the unitary representations, the AdS(1,2) spacetime is suitably parametrized and a representation of SO(1,2) is obtained, from which the Schrödinger equation with Poschl-Teller potential is immediately deduced. Finally, we discuss some relations between the relativistic harmonic oscillator and the Klein-Gordon equation, using the AdS(1,2) static frame. Possible applications of the presented formalism are provided.

Keywords : Schrödinger equation; Pöschl-Teller potential; Casimir; spin and ladder operators; Cartan form; unitary representations; anti-de Sitter spacetime; hyperbolical coordinates; quantum mechanics.

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