No abstract

An algebraic treatment of shape-invariant potentials in supersymmetric quantum mechanics is discussed. By introducing an operator which reparametrizes wave functions, the shape-invariance condition can be related to a oscillator-like algebra. It makes possible to define a coherent state associated with the shape-invariant potentials. For a large class of such potentials, it is shown that the introduced coherent state has the property of resolution of unity.Comment: 11 pages + 1 figure (not included),Plain Tex YITP/K-1019, RCNP-05

The first-order differential Lévy-Leblond equations (LLE's) are the non-relativistic analogs of the Dirac equation, being square roots of (1 + d)-dimensional Schrödinger or heat equations. Just like the Dirac equation, the LLE's possess a natural supersymmetry. In previous works it was shown that non supersymmetric PDE's (notably, the Schrödinger equations for free particles or in the presence of a harmonic potential), admit a natural Z 2 -graded Lie symmetry.In this paper we show that, for a certain class of supersymmetric PDE's, a natural Z 2 ×Z 2graded Lie symmetry appears. In particular, we exhaustively investigate the symmetries of the (1 + 1)-dimensional Lévy-Leblond Equations, both in the free case and for the harmonic potential. In the free case a Z 2 × Z 2 -graded Lie superalgebra, realized by first and secondorder differential symmetry operators, is found. In the presence of a non-vanishing quadratic potential, the Schrödinger invariance is maintained, while the Z 2 -and Z 2 × Z 2 -graded extensions are no longer allowed.The construction of the Z 2 × Z 2 -graded Lie symmetry of the (1 + 2)-dimensional free heat LLE introduces a new feature, explaining the existence of first-order differential symmetry operators not entering the super Schrödinger algebra. The fact that a Lie superalgebra appears even in a purely bosonic setting is not so surprising. Indeed, for the harmonic oscillator, the old results of [20] can be expressed, in modern language, by stating that the Fock vacuum of creation/annihilation operators can be replaced by a lowest weight representation of an osp(1|2) spectrum-generating superalgebra.

It is shown that the N=1 supersymmetric quantum mechanics (SQM) can be extended to a Z2n-graded superalgebra. This is done by presenting quantum mechanical models that realize, with the aid of Clifford gamma matrices, the Z2n-graded Poincaré algebra in one-dimensional spacetime. Reflecting the fact that the Z2n-graded Poincaré algebra has a number of central elements, a sequence of models defining the Z2n-graded version of SQM is provided for a given value of n. In a model of the sequence, the central elements having the same Z2n-degree are realized as dependent or independent operators. It is observed that as the Clifford algebras of larger dimension are used, more central elements are realized as independent operators.

ℓ-Conformal Galilei algebra, denoted by g ℓ (d), is a non-semisimple Lie algebra specified by a pair of parameters (d, ℓ). The algebra is regarded as a nonrelativistic analogue of the conformal algebra. We derive hierarchies of partial differential equations which have invariance of the group generated by g ℓ (d) with central extension as kinematical symmetry. This is done by developing a representation theory such as Verma modules, singular vectors of g ℓ (d) and vector field representations for d = 1, 2.