## Spectral Anomalies of the Robin Laplacian in Non-Lipschitz Domains

J. Math. Sci. Univ. Tokyo
Vol. 20 (2013), No. 1, Page 27–90.

Nazarov, Sergey A. ; Taskinen, Jari
Spectral Anomalies of the Robin Laplacian in Non-Lipschitz Domains
We consider the spectral Laplace-Robin problem in bounded peak shaped domains of $\Bbb R^n$, $n \geq 2$. In case of a sufficiently sharp peak and "wrong" sign of the Robin coefficients, the spectrum becomes pathological: the residual spectrum covers the whole complex plane, while all complex numbers are eigenvalues of the adjoint problem operator. Our results solve a spectral problem posed by H. Amann and D. Daners.