Spectral properties of the layer potentials on Lipschitz domains

Abstract

We study the invertibility of the operator $ \beta I - K^*$ in $H^{-\alpha} (\partial\Omega),\ 0\leq\alpha\leq1$ for $\beta\in \mathbf{C} \setminus(-\frac12 , \frac12]$ where $K^*$ is a adjoint operator of the double layer potential $K$ related to the Laplace equation and $\Omega$ is a bounded Lipschitz domain in $\mathbf{R}^n$. Consequently, the spectrum on the real line lies in $(-\frac12 , \frac12]$.