1. Jacobi operators
Both are readily verified. Nevertheless, let me remark that 9, 9* are no derivations
since they do not satisfy Leibnitz rule. This very often makes the discrete case
different (and sometimes also harder) from the continuous one. In particular, many
calculations become much messier and formulas longer.
There is much more to say about relations for the difference expressions (1.15)
analogous to the ones for differentiation. We refer the reader to, for instance, [4],
[87], or [147] and return to (1.13).
Associated with r is the eigenvalue problem ru = zu. The appropriate setting
for this eigenvalue problem is the Hilbert space
However, before we can
pursue the investigation of the eigenvalue problem in ^2(Z), we need to consider
the Jacobi difference equation
(1.19) TU = zu, u ef(Z), z e C.
Using a(n) ^ 0 we see that a solution u is uniquely determined by the values u(no)
and u(no + 1) at two consecutive points no, n0 + 1 (you have to work much harder
to obtain the corresponding result for differential equations). It follows, that there
are exactly two linearly independent solutions.
Combining (1.16) and the summation by parts formula yields Green's formula
(1-20) ]T (/(r5) - (rf)g)(j) = Wn(f,g) - Wm-iU,9)
for f,g e f'{1), where we have introduced the (modified) Wronskian
Wn (/, g) = a(n) (f(n)g(n + 1) - g(n)f(n + 1)).
Green's formula will be the key to self-adjointness of the operator associated with
r in the Hilbert space
(cf. Theorem 1.5) and the Wronskian is much more
than a suitable abbreviation as we will show next.
Evaluating (1.20) in the special case where / and g both solve (1.19) (with the
same parameter z) shows that the Wronskian is constant (i.e., does not depend on
n) in this case. (The index n will be omitted in this case.) Moreover, it is nonzero
if and only if / and g are linearly independent.
Since the (linear) space of solutions is two dimensional (as observed above) we
can pick two linearly independent solutions c, s of (1.19) and write any solution u
of (1.19) as a linear combination of these two solutions
(1-22) u(n) = -c(n) - -s{n).
W(c,s) W{c,s)
For this purpose it is convenient to introduce the following fundamental solutions
(1.23) Tc(z,.,n0) = : c ( : , , n
) ,
fulfilling the initial conditions
(1.24) c(*,no,no) = l,
., no) = z s(;
c(z,n{) + l,n
) = 0,
s(z,n0 + l,n
) = 1.
Most of the time the base point no will be unessential and we will choose no
for simplicity. In particular, we agree to omit no whenever it is 0, that is,
(1.25) c(z,n) = c(z,n,0), s(z,n) = s(z, n,0).
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