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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