Degeneration of the spectral gap with negative Robin parameter

TL;DR

This paper demonstrates that the previously conjectured lower bounds on the spectral gap for Robin Laplacians do not hold for negative Robin parameters, showing the gap can become exponentially small.

Contribution

The paper proves that the spectral gap conjecture for negative Robin parameters fails, providing explicit examples where the gap is exponentially small.

Findings

Spectral gap can be exponentially small for negative Robin parameters

Conjecture on lower bounds for spectral gap does not extend to negative parameters

Explicit domain family illustrating the failure of the conjecture

Abstract

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.