6 1. INTRODUCTION

that for 0 suﬃciently small Z(s) has a meromorphic continuation to a domain

Re(s) s0 − δ for some small δ 0 and has a pole in the disk Dδ with centre s0.

In the present work we deal with a class K of obstacles K such that the shapes

of the connected components Ki of K can be arbitrary (as long as they satisfy the

conditions in the beginning of this section). This leads to significant complications

in applying Ikawa’s ideas. Obviously one needs a more general abstract meromor-

phicity theorem (cf. Theorem 2.1. below) dealing with a whole class S of pairs

(f, ω) of functions, not just a single one, and allowing for more general types of

functions

ˆ,

f ˆ ω and ∆.

The study of the individual operators

˜

L

−sf+ω

carried out in Ch. 4 below is

similar to that in [I4], [I5] only that in our case we have to make all estimates

uniform so that they apply (with the same choice of the constants involved) to all

(f, ω) ∈ S.

Ch. 5 provides uniform appriori estimates for the resolvents of

˜

L

−sf+ω

and

˜

L

−s

ˆ+ˆ+∆

f ω ln

. As in Ikawa’s case, we choose s0 = s0(f, ω) ∈ R so that

˜

L

−sf+ω

has a maximal eigenvalue 1 (possibly a multiple one). However in our case it seems

impossible to separate 1 from the rest of the spectrum by the same neighbourhood

for all (f, ω) ∈ S. It is nevertheless possible to choose a rectangle Πα around 1 of

size α = (α1, α2) such that ∂Πα is uniformly away from spec(

˜

L −sf+ω). Moreover,

this rectangle can be chosen so that its size is uniformly bounded from below (and

above), though its particular choice depends on (f, ω) ∈ S.

This allows for a uniform application of some basic facts from perturbation

theory of linear operators carried out in Ch. 6. As a result one finds a constant

δ 0 such that for any (f, ω), and any suﬃciently small 0, if

ˆ

f and ˆ ω are

suﬃciently close to f and ω, then there exists s ∈ C with |s − s0(f, ω)| δ such

that 1 is an eigenvalue of

˜

L

−s

ˆ+ˆ+∆

f ω ln

. An application of Pollicott’s results in

[Po2] completes the proof of the abstract meromorphicity theorem. The latter is

then used in Ch. 7 to complete the proof of Theorem 1.1.

Naturally, as in Ikawa’s case, one needs estimates of curvatures of convex fronts

propagating in ΩK , and this time these have to be uniform for all obstacles K in

the class K considered. Such estimates are sketched in Ch. 8 following generally

speaking arguments of Ikawa [I1], [I2] (see also Burq [Bu], Sj¨ ostrand [Sj1] and

Sinai [Si2]). There is nothing new in Ch. 8 in terms of ideas compared to the

papers just mentioned; our aim here is to give suﬃciently precise estimates and

demonstrate their uniformity in the class K.

It is quite clear from the above that basic knowledge about spectra of trans-

fer operators is much needed below. This sort of knowledge is provided by the

Ruelle-Perron-Frobenius theorem. We state it in Ch. 3 below in a form suﬃciently

comprehensive to cover the needs of the present work. A proof of it is given in

[St2].

Acknowledgements. I am grateful to Johannes Sj¨ ostrand a discussion with

whom prompted the present study. Part of the work on this paper was done in

2001 during my visit to ANU (Canberra) for the Special Program on Spectral and

Scattering Theory. Thanks are due to Andrew Hassell and Alan McIntosh for their

hospitality and support. Special thanks are due to Plamen Stefanov for useful

comments, and to Vesselin Petkov for constant support and encouragement and for

pointing out several errors in the first draft of the paper.