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 the work of Cheng et al. [Comput. Math. Appl. 60(9), 2556–2563 (2010)] and Horváth [Proc. Am. Math. Soc. 131(4), 1215–1224 (2002)] in the Neumann and Dirichlet endpoint cases to the interpolating regime. We also build on the recent work by Andrews, Clutterbuck, and Hauer (arXiv:2002.06900) in the case of convex and symmetric single-well potentials. In particular, we show that the spectral gap is an increasing function of the Robin parameter for symmetric potentials.