On the spectral gap of one-dimensional Schrödinger operators on large intervals

Abstract

We study the effect of non-negative potentials on the spectral gap of one-dimensional Schrödinger operators in the limit of large intervals. We derive upper bounds on the gap for different classes of potentials and show, as a main result, that the spectral gap of a Schrödinger operator with a non-zero and sufficiently fast decaying potential closes strictly faster than the gap of the free Laplacian. We show optimality of this result in some sense and establish a conjecture towards the actual decay rate of the spectral gap.