Minimizing the ground state energy of an electron in a randomly deformed lattice

TL;DR

This paper characterizes the configuration of displacements in a random Schrödinger operator that minimizes the ground state energy, revealing a periodic pattern of dense clusters as optimal arrangement.

Contribution

It introduces a novel spectral theory phenomenon, "bubbles tend to the boundary," to determine optimal potential placements for energy minimization in disordered systems.

Findings

Periodic dense clusters minimize the ground state energy.

Potential placement at domain corners minimizes the Neumann eigenvalue.

Center placement maximizes the Neumann eigenvalue in rectangular domains.

Abstract

We provide a characterization of the spectral minimum for a random Schr\"odinger operator of the form $H=-\Delta + \sum_{i \in \Z^d}q(x-i-\omega_i)$ in $L^2(\R^d)$, where the single site potential $q$ is reflection symmetric, compactly supported in the unit cube centered at 0, and the displacement parameters $\omega_i$ are restricted so that adjacent single site potentials do not overlap. In particular, we show that a minimizing configuration of the displacements is given by a periodic pattern of densest possible $2^d$-clusters of single site potentials. The main tool to prove this is a quite general phenomenon in the spectral theory of Neumann problems, which we dub ``bubbles tend to the boundary.'' How should a given compactly supported potential be placed into a bounded domain so as to minimize or maximize the first Neumann eigenvalue of the Schr\"odinger operator on this domain? For square or rectangular domains and reflection symmetric potentials, we show that the first Neumann eigenvalue is minimized when the potential sits in one of the corners of the domain and is maximized when it sits in the center of the domain. With different methods we also show a corresponding result for smooth strictly convex domains.