The Fundamental Gap for a Class of Schrodinger Operators on Path and Hypercube Graphs
Stephen P. Jordan
We consider the difference between the two lowest eigenvalues (the fundamental gap) of a Schrodinger operator acting on a class of graph Laplacians. In particular, we derive tight bounds for the set of convex potentials acting on the path and hypercube graphs. Our proof begins by simplifying the problem with variational techniques. We then make extensive use of the Hellman-Feynman theorem, the recurrence relations on the first two eigenvectors, and techniques specific to the discrete space to complete the proof. We prove the tight bound for the hypercube graph as a corollary to our path graph results.
Journal of Mathematical Physics
quantum, spectral gap, eigenvalues, graph
The Fundamental Gap for a Class of Schrodinger Operators on Path and Hypercube Graphs, Journal of Mathematical Physics
(Accessed June 2, 2023)