Page 251 - Determinants and Their Applications in Mathematical Physics

236 6. Applications of Determinants in Mathematical Physics

they are exceptional. In general, determinants cannot be evaluated in sim-
ple form. The definition of a determinant as a sum of products of elements
is not, in general, a simple form as it is not, in general, amenable to many
of the processes of analysis, especially repeated differentiation.
There may exist a section of the mathematical community which believes
that if an equation possesses a determinantal solution, then the determinant
must emerge from a matrix like an act of birth, for it cannot materialize
in any other way! This belief has not, so far, been justified. In some cases,
the determinants do indeed emerge from sets of equations and hence, by
implication, from matrices, but in other cases, they arise as nonlinear alge-
braic and differential forms with no mother matrix in sight. However, we
do not exclude the possibility that new methods of solution can be devised
in which every determinant emerges from a matrix.
Where the integer n appears in the equation, as in the Dale and Toda
equations, n or some function of n appears in the solution as the order of
the determinant. Where n does not appear in the equation, it appears in
the solution as the arbitrary order of a determinant.
The equations in this chapter were originally solved by a variety of meth-
ods including the application of the Gelfand–Levitan–Marchenko (GLM)
integral equation of inverse scattering theory, namely
∞
,
K(x, y, t)+ R(x + y, t)+ K(x, z, t)R(y + z, t) dz =0
x
in which the kernel R(u, t) is given and K(x, y, t) is the function to be
determined. However, in this chapter, all solutions are verified by the purely
determinantal techniques established in earlier chapters.

6.2 Brief Historical Notes

In order to demonstrate the extent to which determinants have entered the
field of differential and other equations we now give brief historical notes on
the origins and solutions of these equations. The detailed solutions follow
in later sections.

6.2.1 The Dale Equation
The Dale equation is

2
y y +4n
1 2
2
(y ) = y ,
x 1+ x
where n is a positive integer. This equation arises in the theory of stationary
axisymmetric gravitational fields and is the only nonlinear ordinary equa-
tion to appear in this chapter. It was solved in 1978. Two related equations,