Abstract:
We consider the problem of S.P. Novikov on the description of level lines of quasiperiodic functions on a plane with an arbitrary number of quasiperiods. The Novikov problem is of fundamental importance for a number of areas of mathematics and physics, in particular, the problem with 3 quasiperiods is extremely important in describing galvanomagnetic phenomena in crystals. The case of 3 quasiperiods has been studied in the greatest detail, and there are also deep analytical results for the case of 4 quasiperiods, as well as a number of general results for an arbitrary number of quasiperiods $N$. The cases of $N > 3$ have not been studied as deeply as the case of $N = 3$, at the same time, they are also of great importance in describing wide classes of physical systems. Here we will consider applications of the Novikov problem in the physics of two-dimensional systems, namely, the physics of two-layer atomic systems, optics, the physics of ultracold atoms, the physics of two-dimensional electron systems, etc. The study of the Novikov problem for special classes of potentials arising in such systems is of great importance in describing many properties of these systems (including their spectral properties), and also allows for a more detailed comparison with different models of random potentials on a plane.
