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

+ε

2

−V(x)

]

y=0,D=

d

dx

,

and is the only linear ordinary equation to appear in this chapter.

The solution for arbitraryV(x) is not known, but in a paper published in

1956 on the reflectionless transmission of planewavesthrough dielectrics,

Kay and Moses solved it in the particular case in which

V(x)=− 2 D

2

(logA),

whereAis a certain determinant of arbitrary order whose elements are

functions ofx. The equation which Kay and Moses solved is therefore

[

D

2

+ε

2

+2D

2

(logA)

]

y=0.

6.2.3 The Toda Equations..................

The differential–difference equations

D(Rn) = exp(−Rn− 1 )−exp(−Rn+1),

D

2

(Rn) = 2 exp(−Rn)−exp(−Rn− 1 )−exp(−Rn+1),D=

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

Rn=−logyn,

yn=D(logun)

transform the first-order equation into

D(logyn)=yn+1−yn− 1 (6.2.1)

and then into

D(un)=

unun+1

un− 1

. (6.2.2)