01173nas a2200145 4500008004100000245009200041210007000133260001500203300001100218490000700229520071000236100002000946700002400966856003700990 2014 eng d00aThe Fundamental Gap for a Class of SchrÃ¶dinger Operators on Path and Hypercube Graphs0 aFundamental Gap for a Class of SchrÃ¶dinger Operators on Path and c2014/03/06 a0521040 v553 a 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.
1 aJarret, Michael1 aJordan, Stephen, P. uhttp://arxiv.org/abs/1403.1473v1