Construction of Schrödinger- and q-Schrödinger Invariant Differential Operators

The maximal group of space-time transformations which are automorphisms of the solution variety of the free Schrödinger equation is the Schrödinger group. Its Lie algebra is the (centrally extended) Schrödinger algebra $$\hat S(n)$$ (n) in n space and one time dimensions. In the first part of this article we present a method to reconstruct the free Schrödinger operator from $$\hat S(n)$$ (n). For this purpose we use a method based on the determination of singular vectors of the Verma modules of the latter algebra. The idea of using this method has been proposed for semisimple Lie algebras by Kostant [1] and others. It was only more recently that Dobrev gave a complete description this programme [2]. As $$\hat S(n)$$ (n) is not semisimple, the method cannot be applied directly here. However a closer inspection shows that a generalization to certain nonsemisimple Lie algebras is possible. If certain requirements are fulfilled such a generalization gives in general a family of partial differential equations with invariant solution variety. Throughout this article we will call the corresponding partial differential operators also invariant in this case. For $$\hat S(n)$$ (n) this family contains the Schrödinger operator.