Spectral properties of the simple layer potential type operators (Q2393431)

The exact asymptotical behavior of singular values of the simple layer potential type operators obtained. The main result is as follows. Let \(\Omega \) be a bounded, simply connected domain in \(\mathbb C\) with analytic boundary, and let \(T: L^2 (\partial \Omega )\to L^2 (\Omega )\) be the operator defined by \[ Tf(z)=(2\pi)^{-1} \int _{\partial \Omega }\ln \left|z-\xi \right|f(\xi )\left|d\xi \right| . \] Then \(s_{n} (T)\approx \left(\frac{\left|\partial \Omega \right|}{2\pi n} \right)^{3/2}\), where \(s_n (T)\) denotes singular values of operator \(T\). (Here, \(a_n\approx b_n\) denotes the fact that \(\lim _{n\to \infty} (a_n/b_n)=1\)).