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

TitleThe Fundamental Gap for a Class of Schrödinger Operators on Path and Hypercube Graphs
Publication TypeJournal Article
Year of Publication2014
AuthorsJarret, M, Jordan, SP
JournalJournal of Mathematical Physics
Volume55
Issue5
Pages052104
Date Published2014/03/06
Abstract

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.

URLhttp://arxiv.org/abs/1403.1473v1
DOI10.1063/1.4878120
Short TitleJ. Math. Phys.