Geodesic
A lemniscate as the spectrum of a~perturbed shift
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 29, Tome 282 (2001), pp. 74-91

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The spectrum of the perturbed shift operator $T$, $T\colon f(n)\mapsto f(n+1)+a(n)f(n)$, in $\ell^2(\mathbf Z)$ is considered for $a(n)$ taking a finite set of values. It is proved that if all values of the function $a(n)$ have uniform frequencies on $\mathbf Z$, then the essential part of the spectrum fills a generalized lemniscate. 
@article{ZNSL_2001_282_a6,
     author = {A. P. Kalupin and V. L. Oleinik},
     title = {A lemniscate as the spectrum of a~perturbed shift},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {74--91},
     publisher = {mathdoc},
     volume = {282},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_282_a6/}
}
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A. P. Kalupin; V. L. Oleinik. A lemniscate as the spectrum of a~perturbed shift. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 29, Tome 282 (2001), pp. 74-91. http://geodesic.mathdoc.fr/item/ZNSL_2001_282_a6/