Tight-binding approximation on the lemniscate

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.