Degeneration of the spectral gap with negative Robin parameter

The spectral gap of the Neumann and Dirichlet Laplacians are each known to have a sharp positive lower bound among convex domains of a given diameter. Between these cases, for each positive value of the Robin parameter an analogous sharp lower bound on the spectral gap is conjectured. In this paper we show the extension of this conjecture to negative Robin parameters fails completely, by proving the spectral gap of an explicit family of domains can be exponentially small.