The Fundamental Gap for a Class of Schrödinger Operators on Path and Hypercube Graphs

We consider the difference between the two lowest eigenvalues (the
fundamental gap) of a Schr\"{o}dinger operator acting on a class of graphs. In
particular, we derive tight bounds for the gap of Schr\"{o}dinger operators
with convex potentials acting on the path graph. Additionally, for the
hypercube graph, we derive a tight bound for the gap of Schr\"{o}dinger
operators with convex potentials dependent only upon vertex Hamming weight. Our
proof makes use of tools from the literature of the fundamental gap theorem as
proved in the continuum combined with techniques unique to the discrete case.
We prove the tight bound for the hypercube graph as a corollary to our path
graph results.