James Kennedy: Optimising the fundamental gap of a quantum graph

Abstract:

The gap conjecture, proved about 15 years ago by Andrews and Clutterbuck, asserts that the fundamental eigenvalue gap of a Schrödinger operator on a convex domain of fixed diameter with a convex potential, is minimised in the degenerate limit by the Schrödinger operator on an interval of the same diameter, with constant potential. This generalises a roughly 30-year-old result of Lavine, for Schrödinger operators with convex potentials on intervals.

Here we explore what kinds of results can hold in the setting of Schrödinger operators on compact metric trees (the most natural graph analogue of convex domains, where convex potentials can be defined in a natural way). We show that, in general, lower bounds analogous to those on domains are not possible.

More precisely, if one fixes the diameter of the graphs and an upper bound on the \(L^\infty\)-norm of the potentials, then there is still asequence of graphs whose fundamental gap converges to zero; while even on a fixed graph, one can find a sequence of convex potentials (whose \(L^\infty\)-norm explodes) such that the fundamental gap converges to zero.

However, on a given graph, if one restricts to potentials whose \(L^\infty\)-norm satisfies an a priori bound, then general compactness results based on Helly's theorem allow one to recover minimising and maximising convex potentials. In this case, the minimisers will be piecewise linear, but not constant in general. In fact, the constant potential being a minimiser seems to be a ``rare'' property in some sense, which we will try to make more precise in the talk.

This is based on joint work with Mohammed Ahrami, Zakaria El Allali, and Evans Harrell.