Abstract

We study the infimum of the spectrum, or ground state energy (g.s.e.), of a discrete Schrödinger operator on $𝜃{ℝ}^d$ parametrized by a potential $V:{ℝ}^d\to {ℝ}_{\ge 0}$ and a frequency parameter $𝜃\in (0,1)$. We relate this g.s.e. to that of a corresponding continuous semiclassical Schrödinger operator on ${ℝ}^d$ with parameter $𝜃$, arising from the same choice of potential. We show that:

1. The discrete g.s.e. is at most the continuous one for continuous periodic $V$ and irrational $𝜃$.

2. The opposite inequality holds up to a factor of $1-o(1)$ as $𝜃\to 0$ for sufficiently regular smooth periodic $V$.

3. The opposite inequality holds up to a constant factor for every bounded $V$ and $𝜃$ with the property that discrete and continuous averages of $V$ on fundamental domains of $𝜃{\mathbb{Z}}^d$ are comparable.

Our proofs are elementary and rely on sampling and interpolation to map low-energy functions for the discrete operator on $𝜃{\mathbb{Z}}^d$ to low-energy functions for the continuous operator on ${ℝ}^d$, and vice versa.

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