Tight-binding approximation on the lemniscate
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 258-276
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In this paper, we consider a first order linear homogeneous difference equation with a periodic coefficient and a complex parameter, $f(n+1)+a(n)f(n)=zf(n)$, $n\in\mathbb Z$. The set of stability $s_a$ of the equation is known to coincide with a lemniscate which is determined by the finite set of values of the coefficient $a(n)$.
The function $a(n)$ is composed of a sum of two periodic functions, $a(n)=a_1(n)+a_2(n)$, where $a_1$ is a fixed function and $a_2$ is a sum of shifts of a given finite function. By analogy with the quantum solid state theory, the asymptotic behavior of the set $s_a$ is discussed as the period of the function $a_2$ tends to infinity.
@article{ZNSL_2000_270_a11,
author = {V. L. Oleinik},
title = {Tight-binding approximation on the lemniscate},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {258--276},
publisher = {mathdoc},
volume = {270},
year = {2000},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a11/}
}
V. L. Oleinik. Tight-binding approximation on the lemniscate. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 258-276. http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a11/