Scattering Phases and Density of States for Exterior Domain

Details Scattering Phases and Density of States for Exterior Domain Scattering Phases and Density of States for Exterior Domain

1995-02-04

arxiv.org/abs/chao-dyn/9502002v1

Nonlinear Sciences

Chaotic Dynamics

15 pages, Postscript, A4

Scientific paper

For a bounded open domain $\Omega\in \real^2$ with connected complement and piecewise smooth boundary, we consider the Dirichlet Laplacian $-\DO$ on $\Omega$ and the S-matrix on the complement $\Omega^c$. Using the restriction $A_E$ of $(-\Delta-E)^{-1}$ to the boundary of $\Omega $, we establish that $A_{E_0}^{-1/2}A_EA_{E_0}^{-1/2}-1$ is trace class when $E_0$ is negative and give bounds on the energy dependence of this difference. This allows for precise bounds on the total scattering phase, the definition of a $\zeta$-function, and a Krein spectral formula, which improve similar results found in the literature.

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