On the Spectral Gap of a Discrete Schrödinger Operator on the Path Graph in the Limit of Large Volume

Abstract In this paper we study the spectral gap of a discrete Schrödinger operator defined on the path graph and illustrate an interesting effect which has been described recently in the continuous setting. More explicitly, in the large-volume limit and in the presence of a certain external potential, it is shown that the spectral gap converges to zero strictly faster than it does for the discrete Laplacian. The underlying mechanism is a combination of the increase in volume and an effective degeneracy of the ground state in the limiting regime.