174
7 Some Models of “Quantum Measurement”
we obtain the eigenvalue problem in the form:
Ec 1 (E) = c 2 (E),
(7.3.15a)
Ec m (E) = c m−1 (E) + c m+1 (E), m = 2, 3, . . . N − 1,
(7.3.15b)
Ec N (E) = c N −1 (E).
(7.3.15c)
The equations (7.3.15) lead to
c m (E) = U m−1 (E/2)c 1 (E),
(7.3.16)
where
U m−1 (z) :=
sin(m arccos z)
sin(arccos z)
(7.3.17)
are the Tshebyshev polynomials of the second kind [129, 8.940]. This is seen from
the recurrent relations for U n following from (7.3.15), cf. [36, III.(27)]:
U n+1 (z) = 2z U n (z) − U n−1 (z), U 0 (z) = 1, U 1 (z) = 2z.
The equation (7.3.15c) has now the form
U N (E/2) = 0,
(7.3.18)
which is the secular equation corresponding to our eigenvalue problem. Its solutions
are
E j = 2 cos
jπ
N + 1
, j = 1.2. . . . N ,
(7.3.19)
hence we have the expressions
c m (E j ) =
2
N + 1
1/2
sin
jmπ
N + 1
.
(7.3.20)
We shall need also the following definition:
J
(N )
n (z) :=
i
n
N + 1
N
j=1
exp
−i z cos
jπ
N + 1
cos
n
jπ
N + 1
.
(7.3.21)
This is an integral sum of Sommerfeld integral representation of the Bessel function
J n (z), see also [129, 8.41]: