It is believed that given any positive integer b and any
integer n > 1,
lim
N®¥
rn,Nb,V(x1, ..., xn)
exists and is equal for all suitable functions
V, when the xi are suitably scaled. When n = 1 it has been
shown that given an appropriate function V,
lim
N®¥
r1,Nb,V (x)
exists and
is equal for all b and for scaled xi. Furthermore,
r1b,V (x) follows the "semicircle law" in the
limit as N®¥.
When n = 2 it has been shown for a large class of
functions that
lim
N®¥
r2,N2,V(x1, x2) = 1 -
æ è
sinp(x1 - x2)
p(x1 - x2)
ö ø
2
.
In the case where n = 2 and b is 1 or 4 limits have
been found for V(x) = [(x2)/2], but universality has not
been proved. We want to show universality for arbitrary b
and n when V is an even polynomial, so
V(x) =
m å
k=1
a2kx2k
for some m Î N,ak Î R. We have made some progress in the case
where n = 2 and b is arbitrary.
File translated from
TEX
by
TTH,
version 3.40. On 25 Jul 2003, 08:01.
