Spectrum of the Robin Laplacian with singular boundary conditions

Nicolas Popoff, University of Bordeaux, France

In this talk, I will review various results about the spectrum of the Robin Laplacian with ''attractive'' boundary condition in a bounded domain. As the Robin parameter gets large, the first eigenvalues go to infinity. I will describe the mechanism giving the first orders of the asymptotics when the domain has corners, and give improvements when the domain is regular. This includes the description of the bottom of the spectrum in infinite cones. For 3d cones with regular cross section, we are able to count the number of discrete eigenvalues. Finally, I will consider the case of a variable, vanishing, Robin coefficient. In that case the operator may not be self-adjoint and we are able to compute the indices of deficiency. These results are joint works with Vincent Bruneau, Konstantin Pankrashkin and Sergei Nazarov.