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Let's look at a physical system of a particle in a one dimensional periodic potential $V(x)$. When the potential satisfies the periodicity condition of the form

$$ V(x + n b) = V(x),$$

this leads to the usual nice structure, due to the (discrete) translational symmetry of the Hamiltonian. Using a bit of group theory we end up with Bloch's wave functions and reduced Schrodinger equation for the periodic part.

I am interested in the generalization of this. Let me assume I have a 'quasi-periodic' potential of the form

$$ V(\phi(x)) = V(x), $$

where $\phi(x)$ can be some more general transformation, like e.g. non-constant translations (for each step $n$ the period $b(n)$ changes), or if there are gaps in the periodicity (e.g. two superposed periodicities).

My general question: what is known for such potentials?

To be a bit more concrete, for non-constant translations I could have something like

$$ \phi(x) = \frac{b(0)}{b(n)} \left(x + \sum_{i=1}^n b(i)\right), $$

i.e. the period size of my lattice could change, but if I translate and rescale the lattice the potential $V(x)$ stays the same.

Given that I don't know much of solid-state physics, I would have assumed people have developed quite sophisticated tools to explore such cases. Group theory, self-similarity, renormalization and such things come to mind. Is that so? Any comments, discussion of what is done and known, or even hints in the right direction would be quite helpful.

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    $\begingroup$ There is a paper by the group of Massimo Inguscio where they superimposed two optical lattices and observed how an ultra cold atomic gas behaves. They could demonstrate the so called Anderson localisation. This might be of interest to you. Here the link to an overview: physicstoday.scitation.org/doi/10.1063/1.3206092 $\endgroup$
    – Semoi
    Commented Jan 5, 2022 at 20:10

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I'll address your question as two separate parts.

First part: The term 'Quasi-periodicity' would imply that a function is close to, but not exactly, periodic. However the type of function you're describing, generally, could be very far from periodic. As an example, take the domain $X$ to be a discrete lattice, then take $\phi$ to be some arbitrary permutation of the lattice. Then define $V(x) = V(y)$ for all $x$ and $y$ that are in the same cycle of the permutation. This allows functions that you would likely not consider to be close to periodic. For an even more extreme example, take $\phi(x)=x$, then any function $V$ satisfies the equation.

Second part: For your concrete example, I assume that $n$ and $b(i)$ are given. Then let

$$ \sum_{i=1}^n b(i) = C $$

and

$$ \frac{b(0)}{b(n)} = D $$

Thus the equation reduces to:

$$ \phi(x) = D\left(x + C\right) $$

or just an affine transform of $x$. I'll assume $D>0$. For $D=1$ this is just the normal periodic case with period $C$. For $D\ne1$, you get the fixed point

$$ x_0 = -\frac{DC}{D-1} $$

Which means that in the limit of recursion, $\phi(x) = \phi(x_0) = x_0$ thus $V(x) = V(x_0)$ for all $x$, i.e. your potential must be constant.

In the literature there are a number of ways that quasi-periodic potentials are defined, for example cases where there are small perturbations to the periodic potential, or where there are two competing periodic potentials. No general solution is known for many of these cases and even numerically calculated spectra can be very intricate. See the references in this article for many different examples.

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  • $\begingroup$ Ok, I might have misused the term 'quasi-periodic' terminology. 'Semi-periodic' is another that comes to mind... but it might also already be taken in the field to mean something else. I mention two examples that I was hoping would be enough to sketch what I meant by 'periodic'... the simplest of course being the simple translation by some constant $b$. But I was not really interested in perturbations around the canonical setup, and I was hoping there would be more explicitly solvable (1D) cases besides this canonical one (leading to Bloch states). $\endgroup$
    – z.v.
    Commented Jan 5, 2022 at 21:34
  • $\begingroup$ About the second part: If I take all my $b(i)$ to be the same, say $b$ I get, the same old $\phi(x) = x+ n b$, and thus I'm back in Bloch's theorem scenario. I don't see how I can recover this limit from what you are suggesting. $\endgroup$
    – z.v.
    Commented Jan 5, 2022 at 21:40
  • $\begingroup$ I addressed that case. If all the $b$ are the same, then you get $D = b(0)/b(n) = 1$, this is the usual periodic case. $\endgroup$
    – Al Nejati
    Commented Jan 5, 2022 at 22:11

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