1 2 M ay 1 99 7 The Supersymmetry of Relativistic Pöschl - Teller Systems

We analyze the supersymmetry and the shape invariance of the potentials of the (1+1) relativistic oscillators we have recently proposed [1, 2]. In the general relativity, the geometric models play the role of kinematics, helping us to understand the characteristics of the classical or quantum free motion on a given background. One of the simplest (1+1) geometric models is that of the quantum relativistic harmonic oscillator represented by a free massive scalar field on the anti-de Sitter static background [3, 4]. Recently, we have generalized this model to a family of quantum models the metrics of which depend on a real parameter [1]. We have shown [2] that this family contains a set of models which can be considered as the relativistic correspondents of the usual nonrelativistic Pöschl-Teller (PT) systems [5, 6]. These models have been referred as relativistic PT systems [2]. They have countable discrete energy spectra and the same square integrable energy eigenfunctions as those known from the nonrelativistic quantum mechanics. However, the significance of the parameters as well as the formula of the energy levels are different. An important property of all these models is that they lead to the usual harmonic oscillator in the nonrelativistic limit. These analytically solvable relativistic problems may have similar properties of supersymmetry and shape invariance as those known from the non-relativistic theory [7]. Here, our aim is to analyze these properties, pointing 1