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 integernappears in the equation, as in the Dale and Toda

equations,nor some function ofnappears in the solution as the order of

the determinant. Wherendoes 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)+

∫

∞

x

K(x, z, t)R(y+z, t)dz=0

in which the kernelR(u, t) is given andK(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

(y

′′

)

2

=y

′

(

y

x

)′

(

y+4n

2

1+x

)′

,

wherenis 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,