Spectral gaps of 1-D Robin Schr\"odinger operators with single-well potentials

TL;DR

This paper establishes precise lower bounds on the spectral gap of 1D Robin Schrödinger operators with single-well potentials, extending previous results to the interpolating Robin boundary conditions and showing the spectral gap increases with the Robin parameter for symmetric potentials.

Contribution

It provides sharp lower bounds for spectral gaps of 1D Robin Schrödinger operators with single-well potentials, generalizing previous work to include Robin boundary conditions.

Findings

Spectral gap bounds are sharp and explicit.

Spectral gap increases with Robin parameter for symmetric potentials.

Results extend previous Dirichlet and Neumann cases to Robin conditions.

Abstract

We prove sharp lower bounds on the spectral gap of 1-dimensional Schr\"odinger 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 Horv\'ath 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.