Page 252 - Determinants and Their Applications in Mathematical Physics

6.2 Brief Historical Notes 237

which appear in Section 4.11.4, were solved in 1980. Cosgrove has published
an equation which can be transformed into the Dale equation.

6.2.2 The Kay–Moses Equation

The one-dimensional Schr¨odinger equation, which arises in quantum theory,
is
d
2 2
D + ε − V (x) y =0, D = ,
dx
and is the only linear ordinary equation to appear in this chapter.
The solution for arbitrary V (x) is not known, but in a paper published in
1956 on the reflectionless transmission of plane waves through dielectrics,
Kay and Moses solved it in the particular case in which
2
V (x)= −2D (log A),
where A is a certain determinant of arbitrary order whose elements are
functions of x. The equation which Kay and Moses solved is therefore

2 2 2
D + ε +2D (log A) y =0.

6.2.3 The Toda Equations

The differential–difference equations
D(R n ) = exp(−R n−1 ) − exp(−R n+1 ),
d
2
D (R n ) = 2 exp(−R n ) − exp(−R n−1 ) − exp(−R n+1 ), D = ,
dx
arise in nonlinear lattice theory. The first appeared in 1975 in a paper by
Kac and van Moerbeke and can be regarded as a discrete analog of the
KdV equation (Ablowitz and Segur, 1981). The second is the simplest of
a series of equations introduced by Toda in 1967 and can be regarded as a
second-order development of the first. For convenience, these equations are
referred to as first-order and second-order Toda equations, respectively.
The substitutions
R n = − log y n ,
y n = D(log u n )

transform the first-order equation into
(6.2.1)
D(log y n )= y n+1 − y n−1
and then into
u n u n+1
D(u n )= . (6.2.2)
u n−1