Jeff Baker | Searcharxiv

arXiv2008

Low energy properties of the random displacement model

We study low-energy properties of the random displacement model, a random Schrödinger operator describing an electron in a randomly deformed lattice. All periodic displacement configurations which minimize the bottom of the spectrum are characterized. While this configuration is essentially unique for dimension greater than one, there are infinitely many different minimizing configurations in the one-dimensional case. The latter leads to unusual low energy asymptotics for the integrated density of states of the one-dimensional random displacement model. For symmetric Bernoulli-distributed displacements it has a $1/\log^2$-singularity at the bottom of the spectrum. In particular, it is not Hölder-continuous.

math-ph↗

arXiv2007

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

We provide a characterization of the spectral minimum for a random Schrödinger operator of the form $H=-Δ+ \sum_{i \in \Z^d}q(x-i-ω_i)$ 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 $ω_i$ 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 Schrödinger 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.

math-ph↗