CHAPTER 1
Introduction
Let K be an obstacle in Rn (n 3, n odd), i.e. a compact subset of Rn with
C∞ boundary ∂K such that ΩK = Rn \ K is connected. One of the main objects
of study in the classical scattering theory (by an obstacle) is the so called scattering
matrix S(z) related to the wave equation in R×Ω with Dirichlet boundary condition
on R × Ω. This is a meromorphic operator-valued function
S(z) :
L2(Sn−1)
−→
L2(Sn−1)
with poles (resonances) {λj}j=1

in the half-plane Im(z) 0 (see [LP1], [M2]
or [Z1]). The resonances can also be defined as the poles of the meromorphic
continuation of the cut-off resolvent of the self-adjoint realization in L2(Rn \ K) of
the Laplacian −∆ with Dirichlet boundary conditions.
A variety of problems in scattering theory deal with extracting geometric in-
formation about K from the distribution of the poles {λj}. In what follows we
describe one particular problem of this kind.
The obstacle K is called trapping if there exists an infinitely long bounded
broken geodesic (in the sense of Melrose and Sj¨ostrand [MS]) in the exterior domain
Ω. It follows from results of Lax-Phillips [LP2] (see also Vainberg [Va] and Melrose-
Sj¨ ostrand [MS]) that if K is non-trapping, then {z ∈: 0 Im(z) α} contains
finitely many poles λj for any α 0 (cf. the Epilogue in [LP1] for more precise
information). In the first edition of their monograph Scattering Theory published
in 1967, Lax and Phillips conjectured that for trapping obstacles there should exist
a sequence {λj} of scattering poles such that Imλj 0 as j ∞. However
M. Ikawa [I1] showed that this is not the case when K is a disjoint union of two
strictly convex compact domains with smooth boundaries. It turns out that in this
particular case the scattering matrix has poles approximately at the points

d
+iδ,
k = 0, ±1,±2, . . ., where d is the distance between the two connected components
K1 and K2 of K and δ 0 is a constant depending only on the curvatures of
∂K at the ends of the shortest segment connecting K1 and K2. Substantial new
information concerning the distribution of poles in this case was later given by C.
Gerard [G].
Ikawa modified the initial conjecture of Lax and Phillips in the following way.
Modified Lax-Phillips Conjecture (MLPC): If K is trapping, then there
exists α 0 such that the strip {z : 0 Im(z) α} contains infinitely many
scattering resonances λj .
By now a lot of results have been obtained on distribution of resonances in
various aspects of scattering theory. We refer the reader to the monograph [M2] of
1
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