The Fundamental Gap for a Class of Schr\"{o}dinger Operators on Path and Hypercube Graphs

Understand

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.