Robin counting functions on cuboids in the critical scaling regime
We consider eigenvalue counting functions of Robin Laplace operators on cuboids where the Robin parameter and the spectral cut-off $\lambda$ are coupled. Our main focus is the critical regime, in which the Robin parameter is proportional to $\sqrt{\lambda}$. We obtain a two-term asymptotic expansion for the counting functions in this coupled setting. In the critical regime, the second term in the asymptotic expansion depends non-trivially on the proportionality constant and interpolates continuously between the corresponding second terms for the Dirichlet and Neumann Laplacians. Outside the critical regime, one recovers the corresponding Dirichlet or Neumann asymptotics. We also establish a P\'olya-type inequality for the counting function whenever the ratio of the Robin parameter and $\sqrt{\lambda}$ exceeds a dimension-dependent threshold.