Robin boundary conditions are used in heat conductance theory to interpolate between a perfectly insulating boundary, described by Neumann boundary conditions, and a temperature fixing boundary, described by Dirichlet boundary conditions. I have recently started to explore the statistics of these Robin eigenvalues for planar domains, and the fluctuations of the gaps between the Robin and Neumann spectrum, in part driven by numerical experimentation. In joint work with Igor Wigman and Nadav Yesha, we found interesting connections with questions from number theory and from quantum ergodicity and several tantalizing open problems.
A couple of very recent preprints are:
Zeev Rudnick, Igor Wigman, Nadav Yesha. Differences between Robin and Neumann eigenvalues https://arxiv.org/abs/2008.07400
Zeev Rudnick and Igor Wigman On the Robin spectrum for the hemisphere. https://arxiv.org/abs/2008.12964