### Abstract:

Schrodinger's equation for the hydrogen molecule ion and the Helmholtz
equation are separable in prolate and oblate spheroidal coordinates respectively.
They share the same form of the angular equation. The first task
in deriving the ground state energy of the hydrogen molecule ion, and
in obtaining finite solutions of the Helmholtz equation, is to obtain the
physically allowed values of the separation of variables parameter. The
separation parameter is not known analytically, and since it can only have
certain values, it is an important parameter to quantify. Chapter 2 of this
thesis investigates an exact method of obtaining the separation parameter.
By showing that the angular equation is solvable in terms of confluent Heun
functions, a new method to obtain the separation parameter was obtained.
We showed that the physically allowed values of the separation of variables
parameter are given by the zeros of the Wronskian of two linearly dependent
solutions to the angular equation. Since the Heun functions are implemented
in Maple, this new method allows the separation parameter to be calculated
to unlimited precision.
As Schrodinger's equation for the hydrogen molecule ion is related to
Helmholtz's equation, this warranted investigation of scalar beams. Tightly
focused optical and quantum particle beams are described by exact solutions
of the Helmholtz equation. In Chapter 3 of this thesis we investigate the
applicability of the separable spheroidal solutions of the scalar Helmholtz
equation as physical beam solutions. By requiring a scalar beam solution
to satisfy certain physical constraints, we showed that the oblate spheroidal
wave functions can only represent nonparaxial scalar beams when the angular
function is odd, in terms of the angular variable. This condition ensures the
convergence of integrals of physical quantities over a cross-section of the beam
and allows for the physically necessary discontinuity in phase at z = 0 on
the ellipsoidal surfaces of otherwise constant phase. However, these solutions
were shown to have a discontinuous longitudinal derivative.
Finally, we investigated the scattering of scalar waves by oblate and
prolate spheroids whose symmetry axis is coincident with the direction of the
incident plane wave. We developed a phase shift formulation of scattering
by oblate and prolate spheroids, in parallel with the partial wave theory of
scattering by spherical obstacles. The crucial step was application of a finite
Legendre transform to the Helmholtz equation in spheroidal coordinates.
Analytical results were readily obtained for scattering of Schrodinger particle
waves by impenetrable spheroids and for scattering of sound waves by
acoustically soft spheroids. The advantage of this theory is that it enables
all that can be done for scattering by spherical obstacles to be carried over
to the scattering by spheroids, provided the radial eigenfunctions are known.