On the Ground State Energies of Discrete and Semiclassical Schr\"odinger Operators

Abstract

We study the infimum of the spectrum, or ground state energy (g.s.e.), of a discrete Schr\"odinger operator on $\theta\mathbb{Z}^d$ parameterized by a potential $V:\mathbb{R}^d\rightarrow\mathbb{R}_{\ge 0}$ and a frequency parameter $\theta\in (0,1)$. We relate this g.s.e. to that of a corresponding continuous semiclassical Schr\"odinger 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 $\theta\rightarrow 0$ 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.