Solution of Scattering and Bound-State Problems by Construction of Approximate Dynamical Symmetries

A concept of "approximate dynamical symmetry" is formulated by means of which we are able to produce approximate solutions to scattering and bound-state problems in a purely group-theoretic manner for essentially any isotropic potential and to any desired degree of accuracy. This concept forms a natural generalization to arbitrary potentials of the familiar Runge-Lenz symmetry of the Coulomb problem and the ${\mathrm{SU}}_3$ symmetry of the harmonic oscillator. The method consists in transforming the dynamical Lie algebra, i.e., the smallest algebra generated by the Hamiltonian and a complete set of dynamical variables, which is usually infinite and simple, into a finite, simple Lie algebra as follows. We first transform the dynamical algebra into another infinite Lie algebra, but one which contains a "large" ideal, by a process similar to Inonu-Wigner contraction, the angular momentum acting as the contraction parameter. The factor algebra modulo this ideal turns out to be a finite, simple Lie algebra which still contains some of the dynamical information. The above two-stage process will be called "truncation." We are able to develop a sequence of such truncations leading to successively higher dimensional simple Lie algebras whereby we obtain successively better approximations to energy levels and phase shifts. The solutions obtained involve all powers of the coupling constant, and the $N$th-order approximation is at least as good as the $N$th-order W.K.B. approximation and probably better. The special role of the angular momentum in the contraction process enables us to relate these group-theoretic methods to the Regge formalism in a natural way. In fact we are able, thereby, to produce exact solutions to dispersion relations for Regge trajectories obtained by including an arbitrarily large but finite set of trajectories in the unitarity relation. As an illustration of the methods developed we include a calculation to the first nontrivial order of energy levels of an anharmonic oscillator and the well-depth parameter and phase shifts for a Yukawa potential.