[For part I of this paper cf. Duke Math. J. 63, No. 1, 217-233 (1991; Zbl 0732.35064).]

For $ℒ$ an $n$-dimensional lattice in ${ℝ}^n$, $Q\in L^2({ℝ}^n/ℒ)$ and $\alpha \in {ℝ}^n$; $\text{spec}(Q;\alpha )$ denote the spectrum of the Schrödinger operator $\text{Δ}+Q$ acting on the space of all functions $u\in L_{\text{loc}}^2\left({ℝ}^n\right)$ which satisfy $u(x+d)=e^{2\pi i\alpha d}u\left(x\right)$ for all $d\in ℒ$. The periodic spectrum of $Q$, $\text{spec}(Q;0)$ is the spectrum of $\text{Δ}+Q$ on the torus $L^2({ℝ}^n/ℒ)$. The collection of all $\text{spec}(Q;\alpha )$, $\alpha \in {ℝ}^n$, is called the Floquet spectrum of $Q$. The following two questions

i) Does the periodic spectrum of $Q$ determine the Floquet spectrum of $Q$?

ii) Does the Floquet spectrum, in a generic situation, determine the potential $Q$ up to isometries of ${ℝ}^n/ℒ$?

have been investigated by various authors under one of the following conditions on the lattice:

co. The only isometries of ${ℝ}^n/ℒ$ are the compositions of translations with $\pm \text{Id}$.

cl. whenever $d$ and $d^{\text{'}}\in ℒ$ with $\left|d\right|=|d^{\text{'}}|$, then $d=\pm d^{\text{'}}$.

In this paper, the authors consider potentials of the form $Q\left(x\right)={\left|\delta \right|}^{2}q\left(\delta x\right)$ for some primitive element $\delta$ in the dual lattice ${ℒ}^{*}$ and some functions $q\in L^2(ℝ/Z)$. They say that such a potential is a one-dimensional potential associated to $q$ in the direction $\delta$. The set of potentials which are Floquet isospectral to the one-dimensional potential $Q$ can be determined completely if $q\in C^1(ℝ/Z)$. It is given by $\{\tilde{Q}\left(x\right)=|\delta {|}^2\tilde{q}\left(\delta x\right):\tilde{q}\in {\text{Iso}}_{C^{\text{'}}}\left(q\right)\}$, where ${\text{Iso}}_{C^{\text{'}}}\left(q\right)$ denotes the set of potentials $\tilde{q}\in C^1(ℝ/Z)$ such that $-\frac{d^2}{d{x}^2}+\tilde{q}$ and $-\frac{d^2}{d{x}^2}+q$ have the same periodic spectrum.

First, they give an affirmative answer for question (i) in dimension two, assuming only that the lattice satisfies the condition co, and in dimension three under additional hypotheses. They answer question (i) negatively in dimension four by constructing a lattice $ℒ$ in ${ℝ}^4$ satisfying co and pairs of potentials on ${ℝ}^4/ℒ$ which have the same periodic spectrum but different Floquet spectra.

After my knowledge, these are the first such examples in any dimension. For the affirmative results they assume either that (a) $n=2$, $ℒ$ satisfies co and $q\in H^2({ℝ}^4/ℒ)$ or (b) $n=2$ or 3 and $ℒ$ is a rational lattice (i.e. all lattice elements have rational coordinates) satisfying a mild genericity condition.