SCATTERING PHASES AND DENSITY-OF-STATES FOR EXTERIOR DOMAINS

For a bounded open domain Omega is an element of R(2) with connected c omplement and piecewise smooth boundary, we consider: the Dirichlet La placian -Delta(Omega) on Omega and the S-matrix on the complement Omeg a(c). Using the restriction A(E) of (-Delta - E)(-1) to the boundary o f Omega, we establish that A(E0)(-1/2) A(E) A(E0)(-1/2)-1 is trace cla ss when E(0) is negative and give bounds on Eo the energy dependence o f this difference. This allows for precise bounds on the total scatter ing phase, the definition of a zeta-function, and a Krein spectral for mula, which improve similar results found in the literature.