1. INTRODUCTION 3
The study of the scattering zeta function FD(s) itself seems to be rather difficult
and very few results about it are known. We refer the reader to the works of Petkov
[P] and Naud [N] for information and references in this direction.
In this paper we develop further the methods of Ikawa in [I4], [I5] to deal with
a whole family K of obstacles K of the form (1.1) contained in some fixed ‘large’ ball
and such that the connected components of K have bounded eccentricity and the
distances between their connected components are uniformly bounded from below.
That is, we assume that there exist constants D0 d0 0 and χ0 1 such that:
(1.2) K {x
Rn
: x D0},
(1.3)
κmax
κmin
χ0 ,
where κmin = κmin
(K)
0 and κmax = κmax
(K)
0 are the minimal and maximal normal
curvatures of ∂K, and
(1.4) di,j (K) = dist(Ki, Kj ) d0 for all i = j , i, j = 1, . . . , p .
We also assume that the class of obstacles K under consideration satisfy the fol-
lowing uniform no eclipse condition:
(1.5) dist( Kk, convex hull(Ki Kj ) ) χ1 for all k = i = j = k
for some constant χ1 0. The conditions (1.2), (1.4) and (1.5) imply the existence
of a constant ν0 0 = ν0(D0, d0, χ1) such that for any three points x ∂Ki,
y ∂Kj , z ∂K such that i = j, j = and the segments [x, y] and [y, z] satisfy
the law of reflection at y with respect to ∂K, i.e. if locally near the end point y, the
segments [x, y] and [y, z] are symmetric with respect to the exterior unit normal
νK (y) to ∂K at y, then
(1.6)
z y
z y
, νK (y) ν0 .
Here ., . and . are the standard inner product and norm in Rn.
Set
d(K) = max
i=j
di,j (K) , δ(K) = max
1≤i≤p
diam(Ki) .
Finally we assume that for some given constants γ0 0 and Γ0 0 the obstacles K
under consideration satisfy the following gap condition concerning the distribution
of the numbers di,j (K):
for any i, j = 1, . . . , p either d(K) di,j (K) Γ0
(δ(K))γ0
(1.7)
or d(K) di,j (K) γ0 .
The aim of this work is to prove the following.
Theorem 1.1. For any integer p 3 and any positive constants D0 d0,
χ0 1, χ1, Γ0 and γ0 there exists
0
= 0(p, D0, d0, χ0, χ1, Γ0, γ0) 0 such that
if K is an obstacle of the form (1.1) in
Rn,
where Ki are strictly convex compact
domains in
Rn
with
C∞
boundaries such that diam(Ki)
0
for all i = 1, . . . , p
and K satisfies the conditions (1.2) (1.5) and (1.7), then the MLPC holds for K.
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