1.1. Generai properties 9
Observe that by looking at the Wronskian of two solutions u G
V±(\),
v £
V±(X)
it is not hard to see that the lemma becomes false if a(n) is exponentially
decreasing.
For later use observe that
0 - 1
1 0
(1.44) Sfon,™)-
1
=
^\j$(z,n,m)TJ-\
J
a{m)
(where 3T denotes the transposed matrix of £) and hence
(1.45) |o(m)|||$(z,n,m)- 1 || = |a(n)|||*(z,n,ro)||.
We will exploit this notation later in this monograph but for the moment we
return to our original point of view.
The equation
(1.46) (r -z)f = g
for fixed z G C, g G -^(Z), is referred to as inhomogeneous Jacobi equation.
Its solution can can be completely reduced to the solution of the corresponding
homogeneous Jacobi equation (1.19) as follows. Introduce
( L 4 7 )
, ,_a(z,n,m)
Then the sequence
(1.48)
where
K(z,n,m)= .
a[m)
f(n) = foc(z,n,n0) + fis{z,n,n0)
n
+ ^ * K(z,n,m)g(m),
m~no-{-l
n-l
(1.49)
E * f0)={
J=n0
]P fU)
for n no
0 for n = no
n0 —1
-
^2 f(fi
for n
j=n
satisfies (1.46) and the initial conditions /(no) = /o, /(^o + 1) = /i a s c a n De
checked directly. The summation kernel K(z,n,m) has the following properties:
K(z, n, n) = 0, if (z, n -f 1, n) =
a(n)~1,
K(z, n, m) = —K(z, m, n), and
^(z, m)i(z, n) u(z, n)f (z, m)
(1.50) K(z,n, m)
W(u(z),v(z))
for any pair i^z), ^(Z) of linearly independent solutions of ru = z?x.
Another useful result is the variation of constants formula. It says that if
one solution u of (1.19) with u(n) ^ 0 for all n G Z is known, then a second (linearly
independent, W(u,v) = 1) solution of (1.19) is given by
n - l
(1.51)
v(n) =u(n) Y2
3=no
aU)u(j)u{j + l)'
Previous Page Next Page