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

TL;DR

This paper investigates the relationship between the ground state energies of discrete and semiclassical Schrödinger operators, establishing bounds and equivalences under various conditions using elementary sampling and interpolation techniques.

Contribution

It provides new bounds and relations between discrete and continuous Schrödinger ground state energies, with elementary proofs based on sampling and interpolation.

Findings

Discrete g.s.e. is at most the continuous one for periodic potentials and irrational frequencies.

As frequency approaches zero, the discrete g.s.e. approximates the continuous one up to a small error.

Bounds relate discrete and continuous averages of the potential for general bounded potentials.

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.