Spectral gaps of 1-D Robin Schrödinger operators with single-well potentials

Abstract

We prove sharp lower bounds on the spectral gap of 1-dimensional Schrödinger operators with Robin boundary conditions for each value of the Robin parameter. In particular, our lower bounds apply to single-well potentials with a centered transition point. This result extends work of Cheng et al. and Horváth in the Neumann and Dirichlet endpoint cases to the interpolating regime. We also build on recent work by Andrews, Clutterbuck, and Hauer in the case of convex and symmetric single-well potentials. In particular, we show the spectral gap is an increasing function of the Robin parameter for symmetric potentials.

18 pages, 4 figures