On the Novikov Problem for Superposition of Periodic Potentials

We consider the Novikov problem, namely, the problem of describing the global geometry of level lines of quasiperiodic functions on a plane, for a special class of two-dimensional potentials. Potentials of this class play an important role in the physics of two-dimensional systems and are defined by superpositions of two periodic potentials with the same rotational symmetry. For different orientations of the periods of the original potentials, the resulting potential can have 4 quasiperiodes or be periodic. The main result of the paper is a proof that quasiperiodic potentials of this class can have open level lines at only one energy level. This property brings these potentials closer to random potentials on a plane, as well as to potentials with 3 quasiperiodes possessing “chaotic” level lines. The paper also presents an estimate for the energy interval containing open level lines of periodic potentials arising at “magic” angles of rotation of the original potentials relative to each other.