Optimizing the Fundamental Eigenvalue Gap of Quantum Graphs

TL;DR

This paper investigates how to optimize the spectral gap of Schrödinger operators on metric graphs by characterizing the shape of potentials that minimize or maximize the gap, revealing non-constant optimal potentials and limitations on bounding the gap by graph diameter.

Contribution

It provides explicit descriptions of optimal potentials for spectral gap optimization on metric graphs, showing they are piecewise linear or constant, and highlights limitations in bounding the gap by diameter.

Findings

Optimal potentials are piecewise linear or constant.

Non-constant potentials can be optimal, unlike in classical domains.

No general bounds on the spectral gap based solely on graph diameter.

Abstract

We study the problem of minimizing or maximizing the fundamental spectral gap of Schr\"odinger operators on metric graphs with either a convex potential or a ``single-well'' potential on an appropriate specified subset. (In the case of metric trees, such a subset can be the entire graph.) In the convex case we find that the minimizing and maximizing potentials are piecewise linear with only a finite number of points of non-smoothness, but give examples showing that the optimal potentials need not be constant. This is a significant departure from the usual scenarios on intervals and domains where the constant potential is typically minimizing. In the single-well case we show that the optimal potentials are piecewise constant with a finite number of jumps, and in both cases give an explicit estimate on the number of points of non-smoothness, respectively jumps, the minimizing potential can have. Furthermore, we show that, unlike on domains, it is not generally possible to find nontrivial bounds on the fundamental gap in terms of the diameter of the graph alone, within the given classes.