On the Ground State Energies of Discrete and Semiclassical Schrödinger Operators

Abstract

We study the infimum of the spectrum, or ground state energy (g.s.e.), of a discrete Schrödinger operator on $\theta {\mathbb{Z}}^d$ parameterized by a potential $V:{\mathbb{R}}^d\to {\mathbb{R}}_{\ge 0}$ and a frequency parameter $\theta \in (0,1)$. We relate this g.s.e. to that of a corresponding continuous semiclassical Schrödinger operator on ${\mathbb{R}}^d$ with parameter $\theta$, arising from the same choice of potential. We show that: the discrete g.s.e. is at most the continuous one for continuous periodic $V$ and irrational $\theta$; the opposite inequality holds up to a factor of $1-o(1)$ as $\theta \to 0$ for sufficiently regular smooth periodic $V$; and the opposite inequality holds up to a constant factor for every bounded $V$ and $\theta$ with the property that discrete and continuous averages of $V$ on fundamental domains of $\theta {\mathbb{Z}}^d$ are comparable. Our proofs are elementary and rely on sampling and interpolation to map low-energy functions for the discrete operator on $\theta {\mathbb{Z}}^d$ to low-energy functions for the continuous operator on ${\mathbb{R}}^d$, and vice versa.