A Lie algebraic study of some Schrödinger equations

Bound-state solutions of several different Schrödinger equations are calculated rather efficiently using the methods of Lie algebra. In all cases considered, either of the algebras SO(3) or SO(2,1) provides a suitable framework, and there is no advantage to either choice. However, the choice of an appropriate realization of the generators is of greater significance, leading to a particularly simple solution whenever the Hamiltonian can be expressed as a linear function of the generators. In certain cases, the Hamiltonian can be expressed only as a bilinear function of the generators and only part of the bound-state spectrum can be calculated analytically to yield a finite set of so-called quasiexact solutions. These may be interrelated by means of suitable ladder operators and it is possible, though by no means necessary, to adopt a particular finite-dimensional representation of the underlying algebra. The present work emphasizes the role of similarity transformations, and the connection between the Lie method and other (generally variational) procedures which lead to the diagonalization of very large (in principle infinite-dimensional) matrices.