Path Integral Approach to New Exactly Solvable Systems Without Spherical Symmetry.

Using the path integral approach, the quantum mechanical propagator for the nonspherical systems,(UNFORMATTED TABLE OR EQUATION FOLLOWS)eqalign {rm V(r,theta)&= alpha r ^2 + {betaover r^2sin^2 theta} + {gammaover r^2cos ^2theta},crrm V(r,theta, phi)&= alpha r^2 + {beta over r^2sin^2theta sin^2 phi} + {gammaover r^2sin ^2theta cos^2phi},cr }(TABLE/EQUATION ENDS)is evaluated in polar coordinates. The wavefunctions and energy spectrum are also derived directly from the propagator. The Green's function for the nonspherical systems described by,(UNFORMATTED TABLE OR EQUATION FOLLOWS) eqalign{rm V(r,theta)&= alpha/r + {beta cos thetaover r^2sin^2theta } + {gammaover r^2sin^2 theta},crrm V(r,theta,phi)&= alpha/r + {betaover r^2sin ^2theta sin^2phi} + { gammaover r^2sin^2theta cos^2phi} + {deltaover r^2cos^2theta},cr} (TABLE/EQUATION ENDS)are derived by explicit polar coordinate path integration. The poles and the residues at the poles of the radial Green's functions yield the energy spectrum and the wavefunctions, respectively. The fifth class of potentials considered is of the form,(UNFORMATTED TABLE OR EQUATION FOLLOWS) rm V(r,theta,phi) = { alphaover r^2sin^2theta sin^2phi} + {betaover r^2cos^2theta} + {gamma quad cos phiover r^2sin^2 theta sin^2phi},(TABLE/EQUATION ENDS)which has only scattering states. The propagator is also evaluated by explicit polar coordinate path integration. The angular wavefunctions we obtained for the systems with axial symmetry are expressed in terms of generalized spherical harmonics involving Jacobi polynomials of the polar angle theta and the usual azimuthal dependence. For those which are nonspherically symmetric and explicitly depend on theta and phi, the angular wave functions are generalized zonal functions involving products of Jacobi polynomials of the angles theta and phi. Finally, we stress that, except for certain special cases, the solutions for the above systems have been obtained for the first time in this work.